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Properties

Label 2-650-65.64-c1-0-6
Degree $2$
Conductor $650$
Sign $-0.374 - 0.927i$
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  + 2-s + 2.88i·3-s + 4-s + 2.88i·6-s + 1.88·7-s + 8-s − 5.30·9-s + 6.18i·11-s + 2.88i·12-s + (0.287 − 3.59i)13-s + 1.88·14-s + 16-s − 3i·17-s − 5.30·18-s + 4.88i·19-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.66i·3-s + 0.5·4-s + 1.17i·6-s + 0.711·7-s + 0.353·8-s − 1.76·9-s + 1.86i·11-s + 0.831i·12-s + (0.0798 − 0.996i)13-s + 0.502·14-s + 0.250·16-s − 0.727i·17-s − 1.25·18-s + 1.12i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-0.374 - 0.927i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ -0.374 - 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30553 + 1.93494i\)
\(L(\frac12)\) \(\approx\) \(1.30553 + 1.93494i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 \)
13 \( 1 + (-0.287 + 3.59i)T \)
good3 \( 1 - 2.88iT - 3T^{2} \)
7 \( 1 - 1.88T + 7T^{2} \)
11 \( 1 - 6.18iT - 11T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 - 4.88iT - 19T^{2} \)
23 \( 1 - 0.575iT - 23T^{2} \)
29 \( 1 + 5.07T + 29T^{2} \)
31 \( 1 + 7.30iT - 31T^{2} \)
37 \( 1 - 9.18T + 37T^{2} \)
41 \( 1 + 5.45iT - 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 2.45T + 47T^{2} \)
53 \( 1 + 11.0iT - 53T^{2} \)
59 \( 1 - 8.83iT - 59T^{2} \)
61 \( 1 - 3.64T + 61T^{2} \)
67 \( 1 + 3.57T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 11.8T + 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 - 0.188T + 83T^{2} \)
89 \( 1 - 11.4iT - 89T^{2} \)
97 \( 1 - 15.7T + 97T^{2} \)
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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

−10.82744272643945737009328048524, −9.848842161129824002621944342134, −9.602593367238641476891934660335, −8.150422798565484300700408138983, −7.38300104009298371360766398724, −5.86883678597599177283897669319, −5.05381445290381661694248254872, −4.42252743870259468925208382517, −3.57226023354212662559612392132, −2.21530932399543599259716231944, 1.08971189062211291265923442626, 2.24360739064776733950865404564, 3.43736259913484992816152324403, 4.89612468778068957276601977755, 6.04918693283893376104635808067, 6.51900765395906712510590491937, 7.54758723593737531638824684363, 8.296391429197830717537820266200, 9.046539157734652401898433161593, 10.91025963500120372148951680189

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