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Properties

Label 2-950-19.7-c1-0-30
Degree $2$
Conductor $950$
Sign $-0.823 + 0.567i$
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 + (−1.17 − 2.02i)3-s + (−0.499 + 0.866i)4-s + (1.17 − 2.02i)6-s − 1.34·7-s − 0.999·8-s + (−1.23 + 2.14i)9-s + 3.25·11-s + 2.34·12-s + (−0.745 + 1.29i)13-s + (−0.670 − 1.16i)14-s + (−0.5 − 0.866i)16-s + (−3.29 − 5.70i)17-s − 2.47·18-s + (−1.25 − 4.17i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.675 − 1.17i)3-s + (−0.249 + 0.433i)4-s + (0.477 − 0.827i)6-s − 0.506·7-s − 0.353·8-s + (−0.413 + 0.715i)9-s + 0.982·11-s + 0.675·12-s + (−0.206 + 0.358i)13-s + (−0.179 − 0.310i)14-s + (−0.125 − 0.216i)16-s + (−0.798 − 1.38i)17-s − 0.584·18-s + (−0.288 − 0.957i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.823 + 0.567i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.823 + 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.823 + 0.567i$
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: \((2,\ 950,\ (\ :1/2),\ -0.823 + 0.567i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.156468 - 0.502644i\)
\(L(\frac12)\) \(\approx\) \(0.156468 - 0.502644i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.25 + 4.17i)T \)
good3 \( 1 + (1.17 + 2.02i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 1.34T + 7T^{2} \)
11 \( 1 - 3.25T + 11T^{2} \)
13 \( 1 + (0.745 - 1.29i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.29 + 5.70i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.07 - 1.86i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.65 - 4.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.76T + 31T^{2} \)
37 \( 1 + 5.27T + 37T^{2} \)
41 \( 1 + (2.79 + 4.83i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.81 + 6.61i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.465 + 0.806i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.54 - 7.87i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.0837 - 0.145i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.84 - 10.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.53 + 11.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.59 + 6.22i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.62 + 13.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.75 - 13.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.313T + 83T^{2} \)
89 \( 1 + (-4.90 + 8.49i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.01 - 10.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.314218854422632137736086335610, −8.955170105439866059913354616507, −7.55060992653509944457734587166, −6.90102932533418241182685774559, −6.57387104156464530558290149617, −5.54985079889287640726008489041, −4.61030327890411678957072796908, −3.34131071022528487651731579187, −1.87812978776303437628496931079, −0.22941756732342328381337363808, 1.82650470358016749049197294283, 3.48382618567641783788086731885, 4.05233677148491863526324005773, 4.91922342289937139727142559455, 5.98611176967760087324284798941, 6.50216435518491261012037012450, 8.092083896779400557701038175739, 9.059504878857451510971891097398, 9.918416802664559215861266758101, 10.28842342571434702047272842513

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