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Properties

Label 2-650-65.37-c1-0-11
Degree $2$
Conductor $650$
Sign $0.879 + 0.476i$
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 + (1.71 + 0.460i)3-s + (0.499 − 0.866i)4-s + (1.71 − 0.460i)6-s + (0.386 − 0.670i)7-s − 0.999i·8-s + (0.142 + 0.0825i)9-s + (2.36 + 0.634i)11-s + (1.25 − 1.25i)12-s + (3.22 + 1.61i)13-s − 0.773i·14-s + (−0.5 − 0.866i)16-s + (−0.325 − 1.21i)17-s + 0.165·18-s + (−0.0463 − 0.173i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.992 + 0.265i)3-s + (0.249 − 0.433i)4-s + (0.701 − 0.187i)6-s + (0.146 − 0.253i)7-s − 0.353i·8-s + (0.0476 + 0.0275i)9-s + (0.713 + 0.191i)11-s + (0.363 − 0.363i)12-s + (0.894 + 0.447i)13-s − 0.206i·14-s + (−0.125 − 0.216i)16-s + (−0.0790 − 0.295i)17-s + 0.0389·18-s + (−0.0106 − 0.0397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.476i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 + 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.879 + 0.476i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ 0.879 + 0.476i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.86938 - 0.726925i\)
\(L(\frac12)\) \(\approx\) \(2.86938 - 0.726925i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.22 - 1.61i)T \)
good3 \( 1 + (-1.71 - 0.460i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-0.386 + 0.670i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.36 - 0.634i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.325 + 1.21i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.0463 + 0.173i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.0925 - 0.345i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-0.581 + 0.335i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.01 - 2.01i)T + 31iT^{2} \)
37 \( 1 + (4.27 + 7.41i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.60 - 9.72i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.194 - 0.0521i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + 9.78T + 47T^{2} \)
53 \( 1 + (0.918 - 0.918i)T - 53iT^{2} \)
59 \( 1 + (0.742 - 0.199i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (7.31 - 12.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.894 - 0.516i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.58 + 1.22i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + 12.2iT - 73T^{2} \)
79 \( 1 - 8.67iT - 79T^{2} \)
83 \( 1 + 8.94T + 83T^{2} \)
89 \( 1 + (-3.34 + 12.4i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-9.71 - 5.61i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.52846219679753308948854456688, −9.511280760867449173271735377078, −8.916820891397890833896947883639, −7.988444238908903579113633954432, −6.84560690915142961656309224605, −5.94859995507167822420834938421, −4.59931396675276255858086460395, −3.79014049796353800731780902379, −2.90513798971131687046548768345, −1.55624168399136523899918468595, 1.78475156340784982884983459013, 3.08386555756357171051891180973, 3.85061568400383638643661865273, 5.17094185479308876809962999633, 6.16946089007509571197024387040, 7.04464399032874684142961303169, 8.262356008165738958254477963675, 8.462807567753357011826912594857, 9.529224173297660082533370907458, 10.74213680015641298829070863036

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