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Properties

Label 2-4608-16.5-c1-0-14
Degree $2$
Conductor $4608$
Sign $-0.923 - 0.382i$
Analytic cond. $36.7950$
Root an. cond. $6.06589$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)5-s + 4i·7-s + (2 + 2i)11-s + (−3 + 3i)13-s + (2 − 2i)19-s + 4i·23-s − 3i·25-s + (−3 + 3i)29-s − 8·31-s + (−4 + 4i)35-s + (1 + i)37-s + 8i·41-s + (2 + 2i)43-s + 8·47-s − 9·49-s + ⋯
L(s)  = 1  + (0.447 + 0.447i)5-s + 1.51i·7-s + (0.603 + 0.603i)11-s + (−0.832 + 0.832i)13-s + (0.458 − 0.458i)19-s + 0.834i·23-s − 0.600i·25-s + (−0.557 + 0.557i)29-s − 1.43·31-s + (−0.676 + 0.676i)35-s + (0.164 + 0.164i)37-s + 1.24i·41-s + (0.304 + 0.304i)43-s + 1.16·47-s − 1.28·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4608\)    =    \(2^{9} \cdot 3^{2}\)
Sign: $-0.923 - 0.382i$
Analytic conductor: \(36.7950\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4608} (3457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4608,\ (\ :1/2),\ -0.923 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.516684575\)
\(L(\frac12)\) \(\approx\) \(1.516684575\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (-1 - i)T + 5iT^{2} \)
7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 + (-2 - 2i)T + 11iT^{2} \)
13 \( 1 + (3 - 3i)T - 13iT^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + (-2 + 2i)T - 19iT^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + (3 - 3i)T - 29iT^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + (-1 - i)T + 37iT^{2} \)
41 \( 1 - 8iT - 41T^{2} \)
43 \( 1 + (-2 - 2i)T + 43iT^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + (1 + i)T + 53iT^{2} \)
59 \( 1 + (-6 - 6i)T + 59iT^{2} \)
61 \( 1 + (3 - 3i)T - 61iT^{2} \)
67 \( 1 + (-2 + 2i)T - 67iT^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-10 + 10i)T - 83iT^{2} \)
89 \( 1 + 14iT - 89T^{2} \)
97 \( 1 + 16T + 97T^{2} \)
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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

−8.979517813796907430883645491759, −7.81018731509377498457742598454, −7.13382360727679963366358994625, −6.44840267835882192432814507882, −5.70912287905670638908115558475, −5.07573528750851677650288197188, −4.20133515384481671265121458018, −3.08189567173313599820962520356, −2.31199355680156840744563380044, −1.64132683647081025443506221245, 0.41491015004468414601844303693, 1.27071299617617662536719783295, 2.43617940295902289040837478800, 3.67239753499002969861231361904, 4.02709059208318173774192784167, 5.19822272498312438838063789690, 5.63467336793547198346468266805, 6.67084466777357941644165444161, 7.34148310313545833153577936953, 7.85313224915706363771166654179

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