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Properties

Label 2-950-25.11-c1-0-9
Degree $2$
Conductor $950$
Sign $0.327 - 0.944i$
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  + (−0.809 + 0.587i)2-s + (0.142 − 0.438i)3-s + (0.309 − 0.951i)4-s + (−2.07 − 0.843i)5-s + (0.142 + 0.438i)6-s + 3.64·7-s + (0.309 + 0.951i)8-s + (2.25 + 1.63i)9-s + (2.17 − 0.535i)10-s + (−4.35 + 3.16i)11-s + (−0.373 − 0.271i)12-s + (−1.25 − 0.914i)13-s + (−2.94 + 2.14i)14-s + (−0.665 + 0.788i)15-s + (−0.809 − 0.587i)16-s + (1.21 + 3.75i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.0823 − 0.253i)3-s + (0.154 − 0.475i)4-s + (−0.926 − 0.377i)5-s + (0.0582 + 0.179i)6-s + 1.37·7-s + (0.109 + 0.336i)8-s + (0.751 + 0.546i)9-s + (0.686 − 0.169i)10-s + (−1.31 + 0.954i)11-s + (−0.107 − 0.0783i)12-s + (−0.349 − 0.253i)13-s + (−0.787 + 0.572i)14-s + (−0.171 + 0.203i)15-s + (−0.202 − 0.146i)16-s + (0.295 + 0.909i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.327 - 0.944i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (761, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.327 - 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.842401 + 0.599613i\)
\(L(\frac12)\) \(\approx\) \(0.842401 + 0.599613i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (2.07 + 0.843i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
good3 \( 1 + (-0.142 + 0.438i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 - 3.64T + 7T^{2} \)
11 \( 1 + (4.35 - 3.16i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.25 + 0.914i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.21 - 3.75i)T + (-13.7 + 9.99i)T^{2} \)
23 \( 1 + (3.02 - 2.20i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.73 + 5.33i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.01 - 3.13i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.55 + 1.12i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-7.84 - 5.70i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 11.5T + 43T^{2} \)
47 \( 1 + (2.68 - 8.24i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.34 - 10.2i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.86 - 3.53i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (10.1 - 7.40i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-2.02 - 6.24i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-1.85 + 5.72i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (2.59 - 1.88i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-4.67 + 14.3i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.437 + 1.34i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-13.0 + 9.49i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-0.702 + 2.16i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.37632422248274657879500090849, −9.219939237404288549773005156906, −8.121071434705465588626894010450, −7.66682003859289104548210911633, −7.48671182459732553405084844164, −5.89836313060119034395772246916, −4.77198573293292328929533238191, −4.40350482055061282611970085496, −2.43494007790560906841375544969, −1.30405549112262629463010064986, 0.64931534871098794592271257397, 2.30058439697406406239195937118, 3.42303044328258165705851705540, 4.41075357174859038540970654936, 5.29473447660936504366999519620, 6.78806620015641418335229492947, 7.71273425498337333362340544959, 8.089146426235113074720436071183, 8.986721704119516847976377292393, 10.02458693446495341661913192492

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