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Properties

Label 2-145-145.17-c1-0-3
Degree $2$
Conductor $145$
Sign $0.969 + 0.243i$
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  − 2.73·2-s − 1.63i·3-s + 5.47·4-s + (−0.903 + 2.04i)5-s + 4.46i·6-s + (1.05 + 1.05i)7-s − 9.51·8-s + 0.337·9-s + (2.46 − 5.59i)10-s + (1.15 − 1.15i)11-s − 8.94i·12-s + (1.53 + 1.53i)13-s + (−2.89 − 2.89i)14-s + (3.33 + 1.47i)15-s + 15.0·16-s + 7.16·17-s + ⋯
L(s)  = 1  − 1.93·2-s − 0.942i·3-s + 2.73·4-s + (−0.403 + 0.914i)5-s + 1.82i·6-s + (0.400 + 0.400i)7-s − 3.36·8-s + 0.112·9-s + (0.781 − 1.76i)10-s + (0.348 − 0.348i)11-s − 2.58i·12-s + (0.424 + 0.424i)13-s + (−0.773 − 0.773i)14-s + (0.861 + 0.380i)15-s + 3.76·16-s + 1.73·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.969 + 0.243i$
Analytic conductor: \(1.15783\)
Root analytic conductor: \(1.07602\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 145,\ (\ :1/2),\ 0.969 + 0.243i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.517861 - 0.0640232i\)
\(L(\frac12)\) \(\approx\) \(0.517861 - 0.0640232i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.903 - 2.04i)T \)
29 \( 1 + (1.03 - 5.28i)T \)
good2 \( 1 + 2.73T + 2T^{2} \)
3 \( 1 + 1.63iT - 3T^{2} \)
7 \( 1 + (-1.05 - 1.05i)T + 7iT^{2} \)
11 \( 1 + (-1.15 + 1.15i)T - 11iT^{2} \)
13 \( 1 + (-1.53 - 1.53i)T + 13iT^{2} \)
17 \( 1 - 7.16T + 17T^{2} \)
19 \( 1 + (1.87 + 1.87i)T + 19iT^{2} \)
23 \( 1 + (-4.16 + 4.16i)T - 23iT^{2} \)
31 \( 1 + (0.504 - 0.504i)T - 31iT^{2} \)
37 \( 1 - 4.74iT - 37T^{2} \)
41 \( 1 + (-2.38 - 2.38i)T + 41iT^{2} \)
43 \( 1 - 8.05iT - 43T^{2} \)
47 \( 1 + 1.86iT - 47T^{2} \)
53 \( 1 + (5.39 - 5.39i)T - 53iT^{2} \)
59 \( 1 - 3.57iT - 59T^{2} \)
61 \( 1 + (0.867 - 0.867i)T - 61iT^{2} \)
67 \( 1 + (-7.09 + 7.09i)T - 67iT^{2} \)
71 \( 1 + 3.30iT - 71T^{2} \)
73 \( 1 + 11.7T + 73T^{2} \)
79 \( 1 + (-2.01 - 2.01i)T + 79iT^{2} \)
83 \( 1 + (5.28 - 5.28i)T - 83iT^{2} \)
89 \( 1 + (10.1 + 10.1i)T + 89iT^{2} \)
97 \( 1 + 6.05iT - 97T^{2} \)
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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

−12.55072336802118855374946002861, −11.66834163868817365135438899384, −10.93068934088555035963142894127, −9.930067753155201469242353237450, −8.687599388429043608740723848615, −7.84579012320735675108727229434, −7.00994057698612605081834699391, −6.22085297379968530752286681713, −2.92472401106778317220535264066, −1.36851071484157471660895127587, 1.27505063027461939698922179655, 3.74061710301242203305646874025, 5.54977714752280221174749831836, 7.34856872373015899847966260754, 8.086686288983822483544675215658, 9.168390323613494844989150491698, 9.848998034887321789359381316490, 10.70554875649521675137253639661, 11.64693255989155245920967652977, 12.66463043948580567544156706045

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