Properties

Label 2-65-65.37-c1-0-0
Degree $2$
Conductor $65$
Sign $0.908 - 0.418i$
Analytic cond. $0.519027$
Root an. cond. $0.720435$
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  + (−0.113 + 0.0656i)2-s + (0.332 + 0.0890i)3-s + (−0.991 + 1.71i)4-s + (2.08 + 0.813i)5-s + (−0.0436 + 0.0116i)6-s + (1.39 − 2.40i)7-s − 0.522i·8-s + (−2.49 − 1.44i)9-s + (−0.290 + 0.0442i)10-s + (−3.91 − 1.04i)11-s + (−0.482 + 0.482i)12-s + (0.756 + 3.52i)13-s + 0.365i·14-s + (0.619 + 0.455i)15-s + (−1.94 − 3.37i)16-s + (−0.627 − 2.34i)17-s + ⋯
L(s)  = 1  + (−0.0804 + 0.0464i)2-s + (0.191 + 0.0513i)3-s + (−0.495 + 0.858i)4-s + (0.931 + 0.363i)5-s + (−0.0178 + 0.00477i)6-s + (0.525 − 0.910i)7-s − 0.184i·8-s + (−0.831 − 0.480i)9-s + (−0.0917 + 0.0140i)10-s + (−1.18 − 0.316i)11-s + (−0.139 + 0.139i)12-s + (0.209 + 0.977i)13-s + 0.0976i·14-s + (0.159 + 0.117i)15-s + (−0.487 − 0.843i)16-s + (−0.152 − 0.567i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.908 - 0.418i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.908 - 0.418i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.881743 + 0.193131i\)
\(L(\frac12)\) \(\approx\) \(0.881743 + 0.193131i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.08 - 0.813i)T \)
13 \( 1 + (-0.756 - 3.52i)T \)
good2 \( 1 + (0.113 - 0.0656i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.332 - 0.0890i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-1.39 + 2.40i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.91 + 1.04i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.627 + 2.34i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.491 - 1.83i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.06 + 7.70i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (3.96 - 2.28i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.87 - 3.87i)T + 31iT^{2} \)
37 \( 1 + (-3.50 - 6.07i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.66 - 6.20i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (6.24 - 1.67i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + 0.512T + 47T^{2} \)
53 \( 1 + (1.32 - 1.32i)T - 53iT^{2} \)
59 \( 1 + (-2.53 + 0.679i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.641 + 1.11i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.13 + 1.80i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.20 - 1.66i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + 9.93iT - 73T^{2} \)
79 \( 1 + 8.37iT - 79T^{2} \)
83 \( 1 - 3.17T + 83T^{2} \)
89 \( 1 + (-1.61 + 6.01i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-10.1 - 5.88i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.60910332682224290837071347407, −13.85561733100869223042742278080, −13.08890944855225626763749194075, −11.58468567024372803098444052686, −10.39700896440475709779908903900, −9.093410446033728386887299520975, −8.035109401672659491852207279265, −6.62466649187397203231335808465, −4.79611633433935499902853179618, −2.98693751433588238010709913889, 2.24766597668038127849594574510, 5.29435403307727805906620731664, 5.64325359802100540988970506774, 8.076563087686087806140083031999, 9.069785455010748420992493004729, 10.18056436398001231297115388852, 11.26296217414437199741886705295, 13.01286097862465430391884531570, 13.62088875811255312394458120220, 14.86158302219870219303896941247

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