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Properties

Label 2-950-19.7-c1-0-6
Degree $2$
Conductor $950$
Sign $-0.761 + 0.648i$
Analytic cond. $7.58578$
Root an. cond. $2.75423$
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 + (1.32 + 2.29i)3-s + (−0.499 + 0.866i)4-s + (−1.32 + 2.29i)6-s − 3.64·7-s − 0.999·8-s + (−2 + 3.46i)9-s − 4.64·11-s − 2.64·12-s + (1 − 1.73i)13-s + (−1.82 − 3.15i)14-s + (−0.5 − 0.866i)16-s − 3.99·18-s + (1.67 + 4.02i)19-s + (−4.82 − 8.35i)21-s + (−2.32 − 4.02i)22-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.763 + 1.32i)3-s + (−0.249 + 0.433i)4-s + (−0.540 + 0.935i)6-s − 1.37·7-s − 0.353·8-s + (−0.666 + 1.15i)9-s − 1.40·11-s − 0.763·12-s + (0.277 − 0.480i)13-s + (−0.487 − 0.843i)14-s + (−0.125 − 0.216i)16-s − 0.942·18-s + (0.384 + 0.923i)19-s + (−1.05 − 1.82i)21-s + (−0.495 − 0.857i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.761 + 0.648i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.761 + 0.648i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.411448 - 1.11728i\)
\(L(\frac12)\) \(\approx\) \(0.411448 - 1.11728i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 + (-1.67 - 4.02i)T \)
good3 \( 1 + (-1.32 - 2.29i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 3.64T + 7T^{2} \)
11 \( 1 + 4.64T + 11T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.822 - 1.42i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.822 - 1.42i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.64T + 31T^{2} \)
37 \( 1 + 0.354T + 37T^{2} \)
41 \( 1 + (-0.145 - 0.252i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.64 - 9.77i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.17 + 3.77i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.29 - 10.8i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.96 - 6.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.468 - 0.811i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.322 + 0.559i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.35 - 2.34i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.854 - 1.47i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.93T + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.85 + 3.21i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.23819852002422515924047330754, −9.708173867512168281264316782727, −8.964981160501977284463752992855, −8.072103321136328039367622791689, −7.31961079333253672065515222228, −6.00318498165610549601956914594, −5.37518293476153348484913917846, −4.25266271722113716231952939406, −3.37591318843800107866642262793, −2.79265073379789577280431490718, 0.41682445812869518206796934422, 2.07623075975983582224591438805, 2.81385448727119162803517107509, 3.67330141053162158537182476336, 5.17473320440158516204700417369, 6.23858454723559367638511949060, 6.99388243013563339167319160144, 7.76413815557263705712815168304, 8.776847915049906449720948629686, 9.459930542145896987201673283650

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