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Properties

Label 2-950-95.64-c1-0-18
Degree $2$
Conductor $950$
Sign $0.702 + 0.711i$
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  + (0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (−0.499 + 0.866i)6-s − 2i·7-s − 0.999i·8-s + (−1 + 1.73i)9-s + 0.999i·12-s + (5.19 + 3i)13-s + (−1 − 1.73i)14-s + (−0.5 − 0.866i)16-s + (6.06 − 3.5i)17-s + 2i·18-s + (−3.5 − 2.59i)19-s + (1 + 1.73i)21-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.204 + 0.353i)6-s − 0.755i·7-s − 0.353i·8-s + (−0.333 + 0.577i)9-s + 0.288i·12-s + (1.44 + 0.832i)13-s + (−0.267 − 0.462i)14-s + (−0.125 − 0.216i)16-s + (1.47 − 0.848i)17-s + 0.471i·18-s + (−0.802 − 0.596i)19-s + (0.218 + 0.377i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.702 + 0.711i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.702 + 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82172 - 0.761061i\)
\(L(\frac12)\) \(\approx\) \(1.82172 - 0.761061i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
19 \( 1 + (3.5 + 2.59i)T \)
good3 \( 1 + (0.866 - 0.5i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (-5.19 - 3i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-6.06 + 3.5i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.73 - i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5 + 8.66i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 + (1 + 1.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-10.3 + 6i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.92 + 4i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6 - 10.3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.59 + 1.5i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 13iT - 83T^{2} \)
89 \( 1 + (6.5 - 11.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12.9 + 7.5i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.26879853961368974037434320458, −9.286269537071185676012766344520, −8.244824071027461324396025992912, −7.25185154559791539516321610789, −6.28187217984361891906604832587, −5.51493464702251269920713037866, −4.51843792320237024018355204104, −3.80482947237208209895949919534, −2.53810334306240301171549339700, −0.978587380227841306439929694807, 1.29976587523019975804123353436, 3.02572344753702151810622451150, 3.77849189061787920699576941870, 5.19950837468394501833989652945, 5.99762977822599129886135023561, 6.26383761212878209321051787158, 7.55519800928388627542491396832, 8.438726879447873201992035378150, 9.033519541571333615044573100231, 10.44314077792612287236862712987

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