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Properties

Label 4-1110e2-1.1-c1e2-0-16
Degree $4$
Conductor $1232100$
Sign $1$
Analytic cond. $78.5597$
Root an. cond. $2.97714$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 5-s + 6-s + 5·7-s − 8-s + 10-s + 4·11-s − 5·13-s + 5·14-s + 15-s − 16-s − 2·17-s + 5·19-s + 5·21-s + 4·22-s − 24-s − 5·26-s − 27-s − 18·29-s + 30-s + 8·31-s + 4·33-s − 2·34-s + 5·35-s + 37-s + 5·38-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.447·5-s + 0.408·6-s + 1.88·7-s − 0.353·8-s + 0.316·10-s + 1.20·11-s − 1.38·13-s + 1.33·14-s + 0.258·15-s − 1/4·16-s − 0.485·17-s + 1.14·19-s + 1.09·21-s + 0.852·22-s − 0.204·24-s − 0.980·26-s − 0.192·27-s − 3.34·29-s + 0.182·30-s + 1.43·31-s + 0.696·33-s − 0.342·34-s + 0.845·35-s + 0.164·37-s + 0.811·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1232100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(78.5597\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1232100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.251390660\)
\(L(\frac12)\) \(\approx\) \(5.251390660\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 - T + T^{2} \)
5$C_2$ \( 1 - T + T^{2} \)
37$C_2$ \( 1 - T + p T^{2} \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.7.af_s
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.f_m
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.17.c_an
19$C_2^2$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.19.af_g
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.29.s_fj
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
41$C_2^2$ \( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.41.ai_x
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.47.ae_du
53$C_2^2$ \( 1 - 8 T + 11 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_l
59$C_2^2$ \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.59.ae_abr
61$C_2$ \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.61.o_ff
67$C_2^2$ \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.67.ac_acl
71$C_2^2$ \( 1 + 9 T + 10 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.71.j_k
73$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.73.abg_pm
79$C_2^2$ \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.79.am_cn
83$C_2^2$ \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.83.aj_ac
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.89.a_adl
97$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.97.i_ic
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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

−9.831343566594205917664549654888, −9.387715447092821216452960063542, −9.350636411152087656773196620287, −8.991774535737113864566293787271, −8.368236106500410927382404112243, −7.78069348500159354035072346839, −7.67836108611901298635361204773, −7.32663558808767185092253249957, −6.67289225518721656527527585044, −6.19557285676697858922175379338, −5.50005982686468226758144248732, −5.40896512500361743712494875341, −4.81426163624664657202230537606, −4.44348654643401444257823562430, −3.91568696558912322202087051379, −3.56268178767474618850704171106, −2.64771732596061523796186109014, −2.22617306963070611371022699356, −1.75047599663205583719936389376, −0.921361052193918108075689182150, 0.921361052193918108075689182150, 1.75047599663205583719936389376, 2.22617306963070611371022699356, 2.64771732596061523796186109014, 3.56268178767474618850704171106, 3.91568696558912322202087051379, 4.44348654643401444257823562430, 4.81426163624664657202230537606, 5.40896512500361743712494875341, 5.50005982686468226758144248732, 6.19557285676697858922175379338, 6.67289225518721656527527585044, 7.32663558808767185092253249957, 7.67836108611901298635361204773, 7.78069348500159354035072346839, 8.368236106500410927382404112243, 8.991774535737113864566293787271, 9.350636411152087656773196620287, 9.387715447092821216452960063542, 9.831343566594205917664549654888

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