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Properties

Label 4-6150e2-1.1-c1e2-0-2
Degree $4$
Conductor $37822500$
Sign $1$
Analytic cond. $2411.59$
Root an. cond. $7.00770$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·3-s + 3·4-s − 4·6-s + 6·7-s − 4·8-s + 3·9-s − 6·11-s + 6·12-s + 4·13-s − 12·14-s + 5·16-s + 6·17-s − 6·18-s − 6·19-s + 12·21-s + 12·22-s + 4·23-s − 8·24-s − 8·26-s + 4·27-s + 18·28-s − 6·29-s − 6·32-s − 12·33-s − 12·34-s + 9·36-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.63·6-s + 2.26·7-s − 1.41·8-s + 9-s − 1.80·11-s + 1.73·12-s + 1.10·13-s − 3.20·14-s + 5/4·16-s + 1.45·17-s − 1.41·18-s − 1.37·19-s + 2.61·21-s + 2.55·22-s + 0.834·23-s − 1.63·24-s − 1.56·26-s + 0.769·27-s + 3.40·28-s − 1.11·29-s − 1.06·32-s − 2.08·33-s − 2.05·34-s + 3/2·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37822500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(37822500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(2411.59\)
Root analytic conductor: \(7.00770\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 37822500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.280576725\)
\(L(\frac12)\) \(\approx\) \(4.280576725\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
41$C_1$ \( ( 1 + T )^{2} \)
good7$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.7.ag_s
11$C_4$ \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.11.g_ba
13$D_{4}$ \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.13.ae_k
17$D_{4}$ \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.17.ag_bm
19$D_{4}$ \( 1 + 6 T + 42 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.19.g_bq
23$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_be
29$D_{4}$ \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.29.g_w
31$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.31.a_as
37$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.37.a_cc
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$D_{4}$ \( 1 - 2 T + 50 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.47.ac_by
53$D_{4}$ \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.53.ae_dm
59$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.59.ai_cc
61$D_{4}$ \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.61.am_da
67$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_cs
71$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.71.u_ji
73$D_{4}$ \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.73.ai_fm
79$D_{4}$ \( 1 - 4 T + 82 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.79.ae_de
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.83.i_ha
89$D_{4}$ \( 1 - 16 T + 222 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.89.aq_io
97$D_{4}$ \( 1 - 10 T + 214 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.97.ak_ig
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.216697400001565348780499549224, −7.982042988497676148142661054577, −7.66737875677556067789817997152, −7.48343729586253144002854153281, −7.24137524027520928901524438046, −6.67910449072440249200597975082, −6.17791727450396438057770800929, −5.64473820139787149001591811075, −5.43574628188603884144782227527, −5.17880883558508065602475910673, −4.43375909209477974261826726920, −4.33914712919037728486890601081, −3.64106138020897022020093211430, −3.33772475273336627924476462217, −2.74261026138627008923962796530, −2.34195390928561585523145891602, −1.92610844858019963774508057329, −1.78580960170885814817352350745, −0.892835671571846767840448227064, −0.78267062911916175287783047166, 0.78267062911916175287783047166, 0.892835671571846767840448227064, 1.78580960170885814817352350745, 1.92610844858019963774508057329, 2.34195390928561585523145891602, 2.74261026138627008923962796530, 3.33772475273336627924476462217, 3.64106138020897022020093211430, 4.33914712919037728486890601081, 4.43375909209477974261826726920, 5.17880883558508065602475910673, 5.43574628188603884144782227527, 5.64473820139787149001591811075, 6.17791727450396438057770800929, 6.67910449072440249200597975082, 7.24137524027520928901524438046, 7.48343729586253144002854153281, 7.66737875677556067789817997152, 7.982042988497676148142661054577, 8.216697400001565348780499549224

Graph of the $Z$-function along the critical line