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Properties

Label 2-650-13.3-c1-0-1
Degree $2$
Conductor $650$
Sign $-0.755 - 0.655i$
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  + (−0.5 − 0.866i)2-s + (1.36 + 2.36i)3-s + (−0.499 + 0.866i)4-s + (1.36 − 2.36i)6-s + (−2.23 + 3.86i)7-s + 0.999·8-s + (−2.23 + 3.86i)9-s + (−0.866 − 1.5i)11-s − 2.73·12-s + (−3.59 + 0.232i)13-s + 4.46·14-s + (−0.5 − 0.866i)16-s + (2.36 − 4.09i)17-s + 4.46·18-s + (−3.59 + 6.23i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.788 + 1.36i)3-s + (−0.249 + 0.433i)4-s + (0.557 − 0.965i)6-s + (−0.843 + 1.46i)7-s + 0.353·8-s + (−0.744 + 1.28i)9-s + (−0.261 − 0.452i)11-s − 0.788·12-s + (−0.997 + 0.0643i)13-s + 1.19·14-s + (−0.125 − 0.216i)16-s + (0.573 − 0.993i)17-s + 1.05·18-s + (−0.825 + 1.42i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-0.755 - 0.655i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ -0.755 - 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.343431 + 0.919954i\)
\(L(\frac12)\) \(\approx\) \(0.343431 + 0.919954i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
13 \( 1 + (3.59 - 0.232i)T \)
good3 \( 1 + (-1.36 - 2.36i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (2.23 - 3.86i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.866 + 1.5i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.36 + 4.09i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.59 - 6.23i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.73 + 3i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.26 + 2.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.73T + 31T^{2} \)
37 \( 1 + (-0.598 - 1.03i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.46 - 6i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.46 - 4.26i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.464T + 47T^{2} \)
53 \( 1 + 1.73T + 53T^{2} \)
59 \( 1 + (4.73 - 8.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.63 - 2.83i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.19 - 10.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.26 - 7.39i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 0.732T + 73T^{2} \)
79 \( 1 - 6.73T + 79T^{2} \)
83 \( 1 - 5.66T + 83T^{2} \)
89 \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.36 - 7.56i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.49144266222012004097235970136, −9.831856307409032872645184183068, −9.440601548266922267162195209131, −8.564060722365837359251142996049, −7.927577955596429008542905619123, −6.24259725220808924826055967879, −5.18643170429196532853531298602, −4.15119722399539605534293757818, −2.97752656888277948434916468307, −2.51531783145462083831709665469, 0.51507192133846132755210390980, 2.01187548205777055911372734968, 3.36452384216245960067513866771, 4.66978377580300877156563387280, 6.24572548106155353088545205998, 6.94836918691122683851685924598, 7.47772488668740152810380700386, 8.137126720931856407990207247855, 9.215353587645191176838525156343, 10.03283118698848312989550640382

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