Properties

Label 2-310464-1.1-c1-0-118
Degree $2$
Conductor $310464$
Sign $1$
Analytic cond. $2479.06$
Root an. cond. $49.7902$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 11-s − 6·13-s − 3·17-s − 5·19-s + 23-s − 25-s − 5·29-s + 8·31-s + 7·37-s − 6·41-s + 11·43-s + 9·47-s + 2·53-s + 2·55-s + 3·59-s + 4·61-s + 12·65-s + 6·67-s − 71-s + 14·73-s + 4·79-s + 16·83-s + 6·85-s + 8·89-s + 10·95-s + 11·97-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.301·11-s − 1.66·13-s − 0.727·17-s − 1.14·19-s + 0.208·23-s − 1/5·25-s − 0.928·29-s + 1.43·31-s + 1.15·37-s − 0.937·41-s + 1.67·43-s + 1.31·47-s + 0.274·53-s + 0.269·55-s + 0.390·59-s + 0.512·61-s + 1.48·65-s + 0.733·67-s − 0.118·71-s + 1.63·73-s + 0.450·79-s + 1.75·83-s + 0.650·85-s + 0.847·89-s + 1.02·95-s + 1.11·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(310464\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(2479.06\)
Root analytic conductor: \(49.7902\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 310464,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.414482738\)
\(L(\frac12)\) \(\approx\) \(1.414482738\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
11 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 - 11 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.43850065522438, −12.25608079768745, −11.85985901248702, −11.19621664954989, −10.95979336409753, −10.42229782605783, −9.884427508376182, −9.482663078627285, −8.978951528438580, −8.422370556483569, −7.996287450661295, −7.501074945674271, −7.260050748596858, −6.571349007914367, −6.199571544042005, −5.521080986052096, −4.875318803414061, −4.623629855309855, −3.991164545129751, −3.682822633936590, −2.774622407744285, −2.316391220408311, −2.059963246021738, −0.8038015755879678, −0.4119441958754088, 0.4119441958754088, 0.8038015755879678, 2.059963246021738, 2.316391220408311, 2.774622407744285, 3.682822633936590, 3.991164545129751, 4.623629855309855, 4.875318803414061, 5.521080986052096, 6.199571544042005, 6.571349007914367, 7.260050748596858, 7.501074945674271, 7.996287450661295, 8.422370556483569, 8.978951528438580, 9.482663078627285, 9.884427508376182, 10.42229782605783, 10.95979336409753, 11.19621664954989, 11.85985901248702, 12.25608079768745, 12.43850065522438

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