Properties

Label 2-950-1.1-c1-0-2
Degree $2$
Conductor $950$
Sign $1$
Analytic cond. $7.58578$
Root an. cond. $2.75423$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 0.519·3-s + 4-s + 0.519·6-s − 4.76·7-s − 8-s − 2.72·9-s + 0.960·11-s − 0.519·12-s − 2.24·13-s + 4.76·14-s + 16-s + 0.249·17-s + 2.72·18-s + 19-s + 2.48·21-s − 0.960·22-s + 9.01·23-s + 0.519·24-s + 2.24·26-s + 2.97·27-s − 4.76·28-s + 6.24·29-s + 2.96·31-s − 32-s − 0.499·33-s − 0.249·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.300·3-s + 0.5·4-s + 0.212·6-s − 1.80·7-s − 0.353·8-s − 0.909·9-s + 0.289·11-s − 0.150·12-s − 0.623·13-s + 1.27·14-s + 0.250·16-s + 0.0605·17-s + 0.643·18-s + 0.229·19-s + 0.541·21-s − 0.204·22-s + 1.88·23-s + 0.106·24-s + 0.441·26-s + 0.573·27-s − 0.901·28-s + 1.16·29-s + 0.531·31-s − 0.176·32-s − 0.0868·33-s − 0.0428·34-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.6304550206\)
\(L(\frac12)\) \(\approx\) \(0.6304550206\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 \)
19 \( 1 - T \)
good3 \( 1 + 0.519T + 3T^{2} \)
7 \( 1 + 4.76T + 7T^{2} \)
11 \( 1 - 0.960T + 11T^{2} \)
13 \( 1 + 2.24T + 13T^{2} \)
17 \( 1 - 0.249T + 17T^{2} \)
23 \( 1 - 9.01T + 23T^{2} \)
29 \( 1 - 6.24T + 29T^{2} \)
31 \( 1 - 2.96T + 31T^{2} \)
37 \( 1 - 0.0399T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 4.96T + 43T^{2} \)
47 \( 1 + 9.49T + 47T^{2} \)
53 \( 1 - 6.84T + 53T^{2} \)
59 \( 1 + 14.5T + 59T^{2} \)
61 \( 1 - 7.53T + 61T^{2} \)
67 \( 1 + 5.72T + 67T^{2} \)
71 \( 1 + 9.61T + 71T^{2} \)
73 \( 1 - 12.0T + 73T^{2} \)
79 \( 1 - 6.07T + 79T^{2} \)
83 \( 1 + 7.45T + 83T^{2} \)
89 \( 1 - 4.07T + 89T^{2} \)
97 \( 1 - 18.4T + 97T^{2} \)
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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

−9.948091539685308432016879974426, −9.234210075517424403643555650970, −8.639001068839464170886260170496, −7.41217508186696347892222880658, −6.62475566274911169556962867734, −6.04700981419646254201216884446, −4.90734770671744497980694283616, −3.30471999333405805959327841466, −2.70461780836463595661692656779, −0.67496779432486798073596660042, 0.67496779432486798073596660042, 2.70461780836463595661692656779, 3.30471999333405805959327841466, 4.90734770671744497980694283616, 6.04700981419646254201216884446, 6.62475566274911169556962867734, 7.41217508186696347892222880658, 8.639001068839464170886260170496, 9.234210075517424403643555650970, 9.948091539685308432016879974426

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