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Properties

Label 2-950-25.6-c1-0-5
Degree $2$
Conductor $950$
Sign $-0.966 + 0.257i$
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  + (0.309 + 0.951i)2-s + (−0.346 + 0.251i)3-s + (−0.809 + 0.587i)4-s + (−1.41 + 1.72i)5-s + (−0.346 − 0.251i)6-s + 1.40·7-s + (−0.809 − 0.587i)8-s + (−0.870 + 2.67i)9-s + (−2.08 − 0.813i)10-s + (1.45 + 4.48i)11-s + (0.132 − 0.407i)12-s + (0.372 − 1.14i)13-s + (0.434 + 1.33i)14-s + (0.0555 − 0.956i)15-s + (0.309 − 0.951i)16-s + (0.893 + 0.648i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.200 + 0.145i)3-s + (−0.404 + 0.293i)4-s + (−0.633 + 0.773i)5-s + (−0.141 − 0.102i)6-s + 0.531·7-s + (−0.286 − 0.207i)8-s + (−0.290 + 0.892i)9-s + (−0.658 − 0.257i)10-s + (0.439 + 1.35i)11-s + (0.0382 − 0.117i)12-s + (0.103 − 0.318i)13-s + (0.116 + 0.357i)14-s + (0.0143 − 0.246i)15-s + (0.0772 − 0.237i)16-s + (0.216 + 0.157i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 + 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.966 + 0.257i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.966 + 0.257i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.131299 - 1.00104i\)
\(L(\frac12)\) \(\approx\) \(0.131299 - 1.00104i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (1.41 - 1.72i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (0.346 - 0.251i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 - 1.40T + 7T^{2} \)
11 \( 1 + (-1.45 - 4.48i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-0.372 + 1.14i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.893 - 0.648i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 + (1.02 + 3.14i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (1.30 - 0.946i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (3.56 + 2.59i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (3.17 - 9.77i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.26 - 6.97i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 12.3T + 43T^{2} \)
47 \( 1 + (-1.14 + 0.829i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-7.56 + 5.49i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.0599 - 0.184i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.937 + 2.88i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-8.09 - 5.88i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-2.61 + 1.90i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.09 + 3.35i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-3.28 + 2.39i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-10.0 - 7.27i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (0.725 + 2.23i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (0.448 - 0.326i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.41512245012818405010315823584, −9.807530347591246546656046775803, −8.422127500302434280689491735551, −7.953331492407613005239183086656, −7.06789301979867818431090343377, −6.39774095182487895336129919352, −5.12173336724568339216840540636, −4.53886306748400918430596808880, −3.43462519594471023439120536272, −2.06307859078798595184657768457, 0.45885649569379646184499972257, 1.63145527061022675969830093115, 3.41907020636611259218087947685, 3.91787789313906307400465245062, 5.19562619871740330363431403567, 5.85346138922661430433213837996, 7.03141656659578820636525127534, 8.193474941442913516081541817245, 8.843785886547008955380146368482, 9.425638946430561216962673609181

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