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Properties

Label 2-650-65.7-c1-0-14
Degree $2$
Conductor $650$
Sign $0.869 + 0.494i$
Analytic cond. $5.19027$
Root an. cond. $2.27821$
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  + (0.866 + 0.5i)2-s + (−0.0901 − 0.336i)3-s + (0.499 + 0.866i)4-s + (0.0901 − 0.336i)6-s + (−0.826 − 1.43i)7-s + 0.999i·8-s + (2.49 − 1.43i)9-s + (−1.55 − 5.78i)11-s + (0.246 − 0.246i)12-s + (−1.83 − 3.10i)13-s − 1.65i·14-s + (−0.5 + 0.866i)16-s + (6.65 + 1.78i)17-s + 2.87·18-s + (5.38 + 1.44i)19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.0520 − 0.194i)3-s + (0.249 + 0.433i)4-s + (0.0368 − 0.137i)6-s + (−0.312 − 0.541i)7-s + 0.353i·8-s + (0.830 − 0.479i)9-s + (−0.467 − 1.74i)11-s + (0.0711 − 0.0711i)12-s + (−0.509 − 0.860i)13-s − 0.442i·14-s + (−0.125 + 0.216i)16-s + (1.61 + 0.432i)17-s + 0.678·18-s + (1.23 + 0.330i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 + 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.869 + 0.494i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ 0.869 + 0.494i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98939 - 0.525957i\)
\(L(\frac12)\) \(\approx\) \(1.98939 - 0.525957i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 \)
13 \( 1 + (1.83 + 3.10i)T \)
good3 \( 1 + (0.0901 + 0.336i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (0.826 + 1.43i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.55 + 5.78i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-6.65 - 1.78i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-5.38 - 1.44i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (4.87 - 1.30i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-3.73 - 2.15i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.72 - 1.72i)T + 31iT^{2} \)
37 \( 1 + (5.19 - 8.99i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.15 + 0.578i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.66 + 6.21i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + 6.02T + 47T^{2} \)
53 \( 1 + (-3.35 + 3.35i)T - 53iT^{2} \)
59 \( 1 + (-1.06 + 3.95i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (4.61 + 7.99i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.59 + 0.919i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.26 - 8.43i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 - 10.0iT - 73T^{2} \)
79 \( 1 - 11.3iT - 79T^{2} \)
83 \( 1 + 9.98T + 83T^{2} \)
89 \( 1 + (3.43 - 0.921i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (7.25 - 4.18i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.25998985709660203644225364378, −9.965761802570515008901099053852, −8.379297047246158019534169542069, −7.78896242179803474723339826918, −6.87859840583699246770827515587, −5.85868248900707577042467172545, −5.21002288411972690479221529704, −3.65766302446672605047280219193, −3.16917030664373383331139461039, −1.01585031924303412499730104949, 1.76440614935904324662141412777, 2.83100809691155360694257387617, 4.27360644655051276840113487354, 4.91518879031258298782725643559, 5.89305440258069847552572449003, 7.23889110550126939879739092600, 7.61431320789258248715487704207, 9.352698759178822550303382314370, 9.856999796679033986012138724602, 10.43527100269996748893708607383

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