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Properties

Label 2-145-5.4-c1-0-3
Degree $2$
Conductor $145$
Sign $0.774 - 0.632i$
Analytic cond. $1.15783$
Root an. cond. $1.07602$
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  − 0.517i·2-s + 1.41i·3-s + 1.73·4-s + (−1.73 + 1.41i)5-s + 0.732·6-s + 2.44i·7-s − 1.93i·8-s + 0.999·9-s + (0.732 + 0.896i)10-s − 1.26·11-s + 2.44i·12-s − 1.79i·13-s + 1.26·14-s + (−2.00 − 2.44i)15-s + 2.46·16-s − 1.41i·17-s + ⋯
L(s)  = 1  − 0.366i·2-s + 0.816i·3-s + 0.866·4-s + (−0.774 + 0.632i)5-s + 0.298·6-s + 0.925i·7-s − 0.683i·8-s + 0.333·9-s + (0.231 + 0.283i)10-s − 0.382·11-s + 0.707i·12-s − 0.497i·13-s + 0.338·14-s + (−0.516 − 0.632i)15-s + 0.616·16-s − 0.342i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.774 - 0.632i$
Analytic conductor: \(1.15783\)
Root analytic conductor: \(1.07602\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 145,\ (\ :1/2),\ 0.774 - 0.632i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13622 + 0.404942i\)
\(L(\frac12)\) \(\approx\) \(1.13622 + 0.404942i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.73 - 1.41i)T \)
29 \( 1 + T \)
good2 \( 1 + 0.517iT - 2T^{2} \)
3 \( 1 - 1.41iT - 3T^{2} \)
7 \( 1 - 2.44iT - 7T^{2} \)
11 \( 1 + 1.26T + 11T^{2} \)
13 \( 1 + 1.79iT - 13T^{2} \)
17 \( 1 + 1.41iT - 17T^{2} \)
19 \( 1 - 3.26T + 19T^{2} \)
23 \( 1 + 6.31iT - 23T^{2} \)
31 \( 1 + 8.73T + 31T^{2} \)
37 \( 1 + 9.14iT - 37T^{2} \)
41 \( 1 + 6.92T + 41T^{2} \)
43 \( 1 - 9.14iT - 43T^{2} \)
47 \( 1 + 1.41iT - 47T^{2} \)
53 \( 1 - 5.93iT - 53T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 - 2.92T + 61T^{2} \)
67 \( 1 + 4.24iT - 67T^{2} \)
71 \( 1 - 3.46T + 71T^{2} \)
73 \( 1 + 7.34iT - 73T^{2} \)
79 \( 1 - 4.19T + 79T^{2} \)
83 \( 1 - 10.1iT - 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 - 10.9iT - 97T^{2} \)
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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

−12.81621856028846601184235497876, −12.03596842987881920574184625758, −11.03153182554657791916680008698, −10.43730024555736516081831010560, −9.313986587855006499243221296222, −7.86304731546637330483995578264, −6.83101821932488606825056155629, −5.37612111897921377861926169509, −3.76486703457796415461676740519, −2.60687080788259311182186792278, 1.53644139966111061058609187826, 3.72238825533018394683207077486, 5.36307509771792475674337794604, 6.97971833532055140979309042546, 7.38496452882958715316303547604, 8.353525387776565250596118767649, 9.986390453138161154561843514867, 11.24624817926543248005570560477, 11.95756884018807431017053763139, 12.98006343541619476453980083567

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