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Properties

Label 2-650-13.5-c2-0-42
Degree $2$
Conductor $650$
Sign $-0.962 - 0.272i$
Analytic cond. $17.7112$
Root an. cond. $4.20846$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − i)2-s + 1.50·3-s − 2i·4-s + (1.50 − 1.50i)6-s + (−2.61 − 2.61i)7-s + (−2 − 2i)8-s − 6.73·9-s + (−4.34 − 4.34i)11-s − 3.01i·12-s + (−12.7 + 2.77i)13-s − 5.23·14-s − 4·16-s − 4.25i·17-s + (−6.73 + 6.73i)18-s + (−7.91 + 7.91i)19-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + 0.502·3-s − 0.5i·4-s + (0.251 − 0.251i)6-s + (−0.374 − 0.374i)7-s + (−0.250 − 0.250i)8-s − 0.747·9-s + (−0.395 − 0.395i)11-s − 0.251i·12-s + (−0.976 + 0.213i)13-s − 0.374·14-s − 0.250·16-s − 0.250i·17-s + (−0.373 + 0.373i)18-s + (−0.416 + 0.416i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.272i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-0.962 - 0.272i$
Analytic conductor: \(17.7112\)
Root analytic conductor: \(4.20846\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1),\ -0.962 - 0.272i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7556591082\)
\(L(\frac12)\) \(\approx\) \(0.7556591082\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
5 \( 1 \)
13 \( 1 + (12.7 - 2.77i)T \)
good3 \( 1 - 1.50T + 9T^{2} \)
7 \( 1 + (2.61 + 2.61i)T + 49iT^{2} \)
11 \( 1 + (4.34 + 4.34i)T + 121iT^{2} \)
17 \( 1 + 4.25iT - 289T^{2} \)
19 \( 1 + (7.91 - 7.91i)T - 361iT^{2} \)
23 \( 1 - 9.41iT - 529T^{2} \)
29 \( 1 + 35.8T + 841T^{2} \)
31 \( 1 + (-17.3 + 17.3i)T - 961iT^{2} \)
37 \( 1 + (0.326 + 0.326i)T + 1.36e3iT^{2} \)
41 \( 1 + (-31.8 + 31.8i)T - 1.68e3iT^{2} \)
43 \( 1 - 3.89iT - 1.84e3T^{2} \)
47 \( 1 + (-1.54 - 1.54i)T + 2.20e3iT^{2} \)
53 \( 1 + 60.3T + 2.80e3T^{2} \)
59 \( 1 + (32.8 + 32.8i)T + 3.48e3iT^{2} \)
61 \( 1 - 83.7T + 3.72e3T^{2} \)
67 \( 1 + (53.7 - 53.7i)T - 4.48e3iT^{2} \)
71 \( 1 + (-57.3 + 57.3i)T - 5.04e3iT^{2} \)
73 \( 1 + (66.4 + 66.4i)T + 5.32e3iT^{2} \)
79 \( 1 - 21.6T + 6.24e3T^{2} \)
83 \( 1 + (58.9 - 58.9i)T - 6.88e3iT^{2} \)
89 \( 1 + (-52.0 - 52.0i)T + 7.92e3iT^{2} \)
97 \( 1 + (34.8 - 34.8i)T - 9.40e3iT^{2} \)
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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

−9.857234784808724816823407286336, −9.220623307101746561219923503884, −8.145201770629437137979983258266, −7.26670805418216568960821374531, −6.08425989521802164688736246242, −5.21710684884582242553718502958, −4.01304086012324635976430103973, −3.07043614483240462945477359398, −2.10438061820763009836152987819, −0.19454102264500513273009146535, 2.33772975114000753244482641549, 3.12236884030837837461571060029, 4.43269224349167386467712373239, 5.40514668209683509443239607839, 6.28883203633796207144411556196, 7.33955202406177889370609382511, 8.106719677444047928774344260443, 8.982076909669162136740982565527, 9.751755688165787334946999115299, 10.86432111889186910486307730123

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