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Properties

Label 2-650-25.11-c1-0-19
Degree $2$
Conductor $650$
Sign $-0.397 + 0.917i$
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  + (−0.809 + 0.587i)2-s + (0.866 − 2.66i)3-s + (0.309 − 0.951i)4-s + (−0.246 + 2.22i)5-s + (0.866 + 2.66i)6-s − 0.630·7-s + (0.309 + 0.951i)8-s + (−3.93 − 2.86i)9-s + (−1.10 − 1.94i)10-s + (0.383 − 0.278i)11-s + (−2.26 − 1.64i)12-s + (−0.809 − 0.587i)13-s + (0.510 − 0.370i)14-s + (5.71 + 2.58i)15-s + (−0.809 − 0.587i)16-s + (−2.10 − 6.48i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.500 − 1.54i)3-s + (0.154 − 0.475i)4-s + (−0.110 + 0.993i)5-s + (0.353 + 1.08i)6-s − 0.238·7-s + (0.109 + 0.336i)8-s + (−1.31 − 0.953i)9-s + (−0.350 − 0.614i)10-s + (0.115 − 0.0839i)11-s + (−0.655 − 0.475i)12-s + (−0.224 − 0.163i)13-s + (0.136 − 0.0990i)14-s + (1.47 + 0.667i)15-s + (−0.202 − 0.146i)16-s + (−0.511 − 1.57i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.397 + 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-0.397 + 0.917i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ -0.397 + 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.534573 - 0.814607i\)
\(L(\frac12)\) \(\approx\) \(0.534573 - 0.814607i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (0.246 - 2.22i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (-0.866 + 2.66i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + 0.630T + 7T^{2} \)
11 \( 1 + (-0.383 + 0.278i)T + (3.39 - 10.4i)T^{2} \)
17 \( 1 + (2.10 + 6.48i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.36 + 7.29i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-6.47 + 4.70i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.765 - 2.35i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.477 - 1.46i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.97 - 1.43i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.47 + 1.07i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 6.94T + 43T^{2} \)
47 \( 1 + (-2.54 + 7.82i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.12 + 3.46i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-8.65 - 6.28i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (6.35 - 4.61i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-4.16 - 12.8i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (4.39 - 13.5i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-13.2 + 9.60i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.49 + 4.60i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.39 + 7.35i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-1.72 + 1.25i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-0.660 + 2.03i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.21752787770317595186924726298, −9.058561051261957777259503463309, −8.499292156030898235674204947642, −7.26479469463573937262510040606, −7.00179432026586698376444802757, −6.43493672135728340791391522982, −4.97064648223162545638134707513, −2.97816481716526274101643572875, −2.34143055116335442649044864774, −0.58660342282097515839402208830, 1.78858903584537102654466973505, 3.44555183238081114293405899204, 4.08059931636452177233131224713, 5.03118416410138579541039241162, 6.21533677512832678138501736899, 7.907781282911806933106402281151, 8.455925306949455690349810162865, 9.327681059614869955457809786895, 9.750290272137612180065962406569, 10.61586363304053070926540272943

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