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Properties

Label 2-65-65.63-c1-0-1
Degree $2$
Conductor $65$
Sign $0.997 - 0.0684i$
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  + (0.0656 + 0.113i)2-s + (−0.0890 + 0.332i)3-s + (0.991 − 1.71i)4-s + (0.813 + 2.08i)5-s + (−0.0436 + 0.0116i)6-s + (−2.40 − 1.39i)7-s + 0.522·8-s + (2.49 + 1.44i)9-s + (−0.183 + 0.229i)10-s + (−3.91 − 1.04i)11-s + (0.482 + 0.482i)12-s + (−3.52 + 0.756i)13-s − 0.365i·14-s + (−0.764 + 0.0847i)15-s + (−1.94 − 3.37i)16-s + (−2.34 + 0.627i)17-s + ⋯
L(s)  = 1  + (0.0464 + 0.0804i)2-s + (−0.0513 + 0.191i)3-s + (0.495 − 0.858i)4-s + (0.363 + 0.931i)5-s + (−0.0178 + 0.00477i)6-s + (−0.910 − 0.525i)7-s + 0.184·8-s + (0.831 + 0.480i)9-s + (−0.0580 + 0.0724i)10-s + (−1.18 − 0.316i)11-s + (0.139 + 0.139i)12-s + (−0.977 + 0.209i)13-s − 0.0976i·14-s + (−0.197 + 0.0218i)15-s + (−0.487 − 0.843i)16-s + (−0.567 + 0.152i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.997 - 0.0684i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.997 - 0.0684i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.951240 + 0.0325913i\)
\(L(\frac12)\) \(\approx\) \(0.951240 + 0.0325913i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−15.12612035816196153129846656883, −13.81490271825954257582665184113, −13.01221330443666137171339185649, −11.12771118512357015887358576094, −10.35339460316726716697396674217, −9.680626957893899816298329673248, −7.34595697593429433371432596138, −6.55848660301563590461273353277, −4.99323771938588425181981196769, −2.67666069128132955871973491241, 2.67227528070768145120377167612, 4.73423588539295803175583088722, 6.49639226989859929796112233136, 7.75013034400788045110669190567, 9.116093717273978295523688913760, 10.23674070180197802496907681510, 11.98042135550261928649294955261, 12.84283351050376695377853129518, 13.10023902450040284158332025565, 15.25194806463317763804026331537

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