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Properties

Label 4-4212e2-1.1-c1e2-0-16
Degree $4$
Conductor $17740944$
Sign $1$
Analytic cond. $1131.17$
Root an. cond. $5.79939$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 3·7-s − 11-s + 2·13-s + 8·17-s − 3·23-s − 6·25-s − 9·29-s − 5·31-s + 3·35-s − 15·37-s − 41-s + 6·43-s − 4·47-s − 4·49-s + 2·53-s + 55-s + 16·59-s − 9·61-s − 2·65-s + 7·67-s + 16·71-s − 8·73-s + 3·77-s − 2·79-s − 18·83-s − 8·85-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.13·7-s − 0.301·11-s + 0.554·13-s + 1.94·17-s − 0.625·23-s − 6/5·25-s − 1.67·29-s − 0.898·31-s + 0.507·35-s − 2.46·37-s − 0.156·41-s + 0.914·43-s − 0.583·47-s − 4/7·49-s + 0.274·53-s + 0.134·55-s + 2.08·59-s − 1.15·61-s − 0.248·65-s + 0.855·67-s + 1.89·71-s − 0.936·73-s + 0.341·77-s − 0.225·79-s − 1.97·83-s − 0.867·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17740944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17740944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(17740944\)    =    \(2^{4} \cdot 3^{8} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1131.17\)
Root analytic conductor: \(5.79939\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 17740944,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
13$C_1$ \( ( 1 - T )^{2} \)
good5$D_{4}$ \( 1 + T + 7 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_h
7$D_{4}$ \( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.7.d_n
11$D_{4}$ \( 1 + T + 19 T^{2} + p T^{3} + p^{2} T^{4} \) 2.11.b_t
17$D_{4}$ \( 1 - 8 T + 37 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.17.ai_bl
19$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \) 2.19.a_z
23$D_{4}$ \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.23.d_t
29$D_{4}$ \( 1 + 9 T + 49 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.29.j_bx
31$D_{4}$ \( 1 + 5 T + 39 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.31.f_bn
37$D_{4}$ \( 1 + 15 T + 127 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.37.p_ex
41$D_{4}$ \( 1 + T + 79 T^{2} + p T^{3} + p^{2} T^{4} \) 2.41.b_db
43$D_{4}$ \( 1 - 6 T + p T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.43.ag_br
47$D_{4}$ \( 1 + 4 T + 85 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.47.e_dh
53$D_{4}$ \( 1 - 2 T + 94 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.53.ac_dq
59$D_{4}$ \( 1 - 16 T + 169 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.59.aq_gn
61$D_{4}$ \( 1 + 9 T + p T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.61.j_cj
67$D_{4}$ \( 1 - 7 T + 117 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.67.ah_en
71$D_{4}$ \( 1 - 16 T + 193 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.71.aq_hl
73$D_{4}$ \( 1 + 8 T + 45 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_bt
79$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.79.c_gd
83$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.83.s_jn
89$D_{4}$ \( 1 + T + 19 T^{2} + p T^{3} + p^{2} T^{4} \) 2.89.b_t
97$D_{4}$ \( 1 + 12 T + 178 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.97.m_gw
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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

−8.069567686358007043332913463112, −7.980417427901968487220631667305, −7.39211484094197334146557102347, −7.23480251098536443094184727252, −6.81717893586566511331489717987, −6.41199261848200123533692486822, −5.88631580784320038455847211756, −5.67331722563642081543327063870, −5.21647317045235709310520704883, −5.19435463610605986877220007971, −4.07471998921469241049956491377, −4.00162063618218934632081074055, −3.57437301277384804320749974122, −3.33996364071000595972697818274, −2.84354298840597988584390405433, −2.18711423332048303474777778081, −1.66168686493448599193665905043, −1.16731681798801957133422056420, 0, 0, 1.16731681798801957133422056420, 1.66168686493448599193665905043, 2.18711423332048303474777778081, 2.84354298840597988584390405433, 3.33996364071000595972697818274, 3.57437301277384804320749974122, 4.00162063618218934632081074055, 4.07471998921469241049956491377, 5.19435463610605986877220007971, 5.21647317045235709310520704883, 5.67331722563642081543327063870, 5.88631580784320038455847211756, 6.41199261848200123533692486822, 6.81717893586566511331489717987, 7.23480251098536443094184727252, 7.39211484094197334146557102347, 7.980417427901968487220631667305, 8.069567686358007043332913463112

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