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Properties

Label 2-145-145.144-c1-0-11
Degree $2$
Conductor $145$
Sign $-0.542 + 0.840i$
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  + 2-s − 2.56·3-s − 4-s + (−0.280 − 2.21i)5-s − 2.56·6-s − 3.46i·7-s − 3·8-s + 3.56·9-s + (−0.280 − 2.21i)10-s − 0.972i·11-s + 2.56·12-s + 4.43i·13-s − 3.46i·14-s + (0.719 + 5.68i)15-s − 16-s − 2·17-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.47·3-s − 0.5·4-s + (−0.125 − 0.992i)5-s − 1.04·6-s − 1.30i·7-s − 1.06·8-s + 1.18·9-s + (−0.0887 − 0.701i)10-s − 0.293i·11-s + 0.739·12-s + 1.23i·13-s − 0.925i·14-s + (0.185 + 1.46i)15-s − 0.250·16-s − 0.485·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.542 + 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.542 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $-0.542 + 0.840i$
Analytic conductor: \(1.15783\)
Root analytic conductor: \(1.07602\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 145,\ (\ :1/2),\ -0.542 + 0.840i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.284357 - 0.521790i\)
\(L(\frac12)\) \(\approx\) \(0.284357 - 0.521790i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.280 + 2.21i)T \)
29 \( 1 + (-4.12 - 3.46i)T \)
good2 \( 1 - T + 2T^{2} \)
3 \( 1 + 2.56T + 3T^{2} \)
7 \( 1 + 3.46iT - 7T^{2} \)
11 \( 1 + 0.972iT - 11T^{2} \)
13 \( 1 - 4.43iT - 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + 3.46iT - 23T^{2} \)
31 \( 1 + 7.90iT - 31T^{2} \)
37 \( 1 - 7.12T + 37T^{2} \)
41 \( 1 + 8.87iT - 41T^{2} \)
43 \( 1 - 5.43T + 43T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
53 \( 1 + 4.43iT - 53T^{2} \)
59 \( 1 + 1.12T + 59T^{2} \)
61 \( 1 - 1.94iT - 61T^{2} \)
67 \( 1 - 12.3iT - 67T^{2} \)
71 \( 1 + 2.24T + 71T^{2} \)
73 \( 1 + 7.12T + 73T^{2} \)
79 \( 1 - 14.8iT - 79T^{2} \)
83 \( 1 + 10.3iT - 83T^{2} \)
89 \( 1 + 8.87iT - 89T^{2} \)
97 \( 1 - 8.24T + 97T^{2} \)
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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

−12.82613899410049206639503140625, −11.79365701359136829052983263604, −11.09131888538413100995644265060, −9.799057026236586362811130366651, −8.655837136957002905605725356901, −6.98043752516577185301006267791, −5.90753858714149880635881960624, −4.65542927772928913942823389993, −4.23482001401805972728799863179, −0.57948483866347112329156145828, 3.05314225209909456921490544241, 4.77661296522691771731853788293, 5.77526998599516028280227635690, 6.35900600865415681870294439017, 8.064330824235915064257298429339, 9.603415377311210289407734908116, 10.64137812842595975177262949167, 11.70276802400966372430483100421, 12.28024545849098531653210661479, 13.16767580842412446624849716055

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