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Properties

Label 2-650-25.16-c1-0-8
Degree $2$
Conductor $650$
Sign $0.285 - 0.958i$
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 + (0.402 + 1.24i)3-s + (0.309 + 0.951i)4-s + (−0.529 + 2.17i)5-s + (0.402 − 1.24i)6-s + 1.35·7-s + (0.309 − 0.951i)8-s + (1.05 − 0.763i)9-s + (1.70 − 1.44i)10-s + (3.62 + 2.63i)11-s + (−1.05 + 0.766i)12-s + (−0.809 + 0.587i)13-s + (−1.09 − 0.797i)14-s + (−2.90 + 0.218i)15-s + (−0.809 + 0.587i)16-s + (−0.240 + 0.741i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.232 + 0.716i)3-s + (0.154 + 0.475i)4-s + (−0.236 + 0.971i)5-s + (0.164 − 0.506i)6-s + 0.512·7-s + (0.109 − 0.336i)8-s + (0.350 − 0.254i)9-s + (0.539 − 0.457i)10-s + (1.09 + 0.793i)11-s + (−0.304 + 0.221i)12-s + (−0.224 + 0.163i)13-s + (−0.293 − 0.213i)14-s + (−0.750 + 0.0563i)15-s + (−0.202 + 0.146i)16-s + (−0.0584 + 0.179i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.285 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.285 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.285 - 0.958i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ 0.285 - 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04008 + 0.775029i\)
\(L(\frac12)\) \(\approx\) \(1.04008 + 0.775029i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.529 - 2.17i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
good3 \( 1 + (-0.402 - 1.24i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 - 1.35T + 7T^{2} \)
11 \( 1 + (-3.62 - 2.63i)T + (3.39 + 10.4i)T^{2} \)
17 \( 1 + (0.240 - 0.741i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.27 + 3.93i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.237 - 0.172i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.37 - 4.23i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.97 - 6.09i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.36 - 0.994i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.23 + 0.899i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 11.9T + 43T^{2} \)
47 \( 1 + (-0.940 - 2.89i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.257 + 0.792i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-8.79 + 6.39i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-8.57 - 6.22i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (0.453 - 1.39i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (0.0974 + 0.299i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-3.32 - 2.41i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.52 + 4.69i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.124 - 0.382i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (1.61 + 1.17i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (3.59 + 11.0i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.57642103120635733816169729897, −9.900921657514684518694773531187, −9.208587027659313941559561900502, −8.337736151683470135901953217201, −7.03288464619898129622429360686, −6.78417042658217110104790821109, −4.93097988068201029420434752343, −3.97697569793222322221464496360, −3.08522632947596676965219227360, −1.65255369882136851953450475392, 0.922842975604042607017948965169, 1.94235512731990928251427232941, 3.85173633094992707023824868977, 4.99582476538459418546829792206, 6.01986996700109628583826365175, 7.01877375997737878392524859383, 8.011291049296932953212940013096, 8.331577182470067765428481209170, 9.327360414547896832387589000913, 10.13221310112092376436271237055

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