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Properties

Label 2-650-65.44-c2-0-31
Degree $2$
Conductor $650$
Sign $-0.0277 + 0.999i$
Analytic cond. $17.7112$
Root an. cond. $4.20846$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + 0.449i·3-s + 2i·4-s + (0.449 − 0.449i)6-s + (6.67 − 6.67i)7-s + (2 − 2i)8-s + 8.79·9-s + (−10.0 − 10.0i)11-s − 0.898·12-s + 13·13-s − 13.3·14-s − 4·16-s − 11.6·17-s + (−8.79 − 8.79i)18-s + (−6 + 6i)19-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + 0.149i·3-s + 0.5i·4-s + (0.0749 − 0.0749i)6-s + (0.953 − 0.953i)7-s + (0.250 − 0.250i)8-s + 0.977·9-s + (−0.911 − 0.911i)11-s − 0.0749·12-s + 13-s − 0.953·14-s − 0.250·16-s − 0.688·17-s + (−0.488 − 0.488i)18-s + (−0.315 + 0.315i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0277 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0277 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-0.0277 + 0.999i$
Analytic conductor: \(17.7112\)
Root analytic conductor: \(4.20846\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1),\ -0.0277 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.525025071\)
\(L(\frac12)\) \(\approx\) \(1.525025071\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
5 \( 1 \)
13 \( 1 - 13T \)
good3 \( 1 - 0.449iT - 9T^{2} \)
7 \( 1 + (-6.67 + 6.67i)T - 49iT^{2} \)
11 \( 1 + (10.0 + 10.0i)T + 121iT^{2} \)
17 \( 1 + 11.6T + 289T^{2} \)
19 \( 1 + (6 - 6i)T - 361iT^{2} \)
23 \( 1 - 33.3T + 529T^{2} \)
29 \( 1 + 48.3T + 841T^{2} \)
31 \( 1 + (-11.3 + 11.3i)T - 961iT^{2} \)
37 \( 1 + (31.6 - 31.6i)T - 1.36e3iT^{2} \)
41 \( 1 + (-45.3 + 45.3i)T - 1.68e3iT^{2} \)
43 \( 1 - 15.9T + 1.84e3T^{2} \)
47 \( 1 + (-59.4 + 59.4i)T - 2.20e3iT^{2} \)
53 \( 1 + 79.6iT - 2.80e3T^{2} \)
59 \( 1 + (53.4 + 53.4i)T + 3.48e3iT^{2} \)
61 \( 1 + 69.0T + 3.72e3T^{2} \)
67 \( 1 + (-58.7 - 58.7i)T + 4.48e3iT^{2} \)
71 \( 1 + (44.7 - 44.7i)T - 5.04e3iT^{2} \)
73 \( 1 + (-67.7 + 67.7i)T - 5.32e3iT^{2} \)
79 \( 1 - 74.7T + 6.24e3T^{2} \)
83 \( 1 + (-5.41 - 5.41i)T + 6.88e3iT^{2} \)
89 \( 1 + (35.6 + 35.6i)T + 7.92e3iT^{2} \)
97 \( 1 + (61.3 + 61.3i)T + 9.40e3iT^{2} \)
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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

−10.46357387100546344881027119377, −9.261444021939980781292460732957, −8.431593128176849108325415470379, −7.64951685262845192862345405531, −6.84538033597799480299200491229, −5.42182577592978879724785581316, −4.31962325895068371636294695699, −3.48255051982399812458445132650, −1.89103383664320379668632683851, −0.72814288258402843999809142284, 1.38878928322988115976235505236, 2.44534624813439080510806869814, 4.35123709983302700715876781933, 5.15617924015385204166939833293, 6.15420136432246860445897393974, 7.27973989044219346570050027279, 7.81459280894043308488669264577, 8.925724189008207082498920707738, 9.359785318210104869227609348585, 10.81790349524117778842732044708

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