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Properties

Label 2-950-25.11-c1-0-42
Degree $2$
Conductor $950$
Sign $-0.742 - 0.669i$
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 + (0.739 − 2.27i)3-s + (0.309 − 0.951i)4-s + (−2.19 + 0.441i)5-s + (−0.739 − 2.27i)6-s − 0.427·7-s + (−0.309 − 0.951i)8-s + (−2.20 − 1.60i)9-s + (−1.51 + 1.64i)10-s + (−4.15 + 3.01i)11-s + (−1.93 − 1.40i)12-s + (−3.25 − 2.36i)13-s + (−0.345 + 0.251i)14-s + (−0.616 + 5.31i)15-s + (−0.809 − 0.587i)16-s + (−0.526 − 1.61i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.427 − 1.31i)3-s + (0.154 − 0.475i)4-s + (−0.980 + 0.197i)5-s + (−0.302 − 0.929i)6-s − 0.161·7-s + (−0.109 − 0.336i)8-s + (−0.736 − 0.535i)9-s + (−0.478 + 0.520i)10-s + (−1.25 + 0.909i)11-s + (−0.559 − 0.406i)12-s + (−0.903 − 0.656i)13-s + (−0.0923 + 0.0671i)14-s + (−0.159 + 1.37i)15-s + (−0.202 − 0.146i)16-s + (−0.127 − 0.392i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 - 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.742 - 0.669i$
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: \((2,\ 950,\ (\ :1/2),\ -0.742 - 0.669i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.303335 + 0.789570i\)
\(L(\frac12)\) \(\approx\) \(0.303335 + 0.789570i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.19 - 0.441i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
good3 \( 1 + (-0.739 + 2.27i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + 0.427T + 7T^{2} \)
11 \( 1 + (4.15 - 3.01i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (3.25 + 2.36i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.526 + 1.61i)T + (-13.7 + 9.99i)T^{2} \)
23 \( 1 + (7.71 - 5.60i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-2.83 + 8.73i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.909 + 2.79i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-2.34 - 1.70i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (7.46 + 5.42i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 10.1T + 43T^{2} \)
47 \( 1 + (-3.83 + 11.8i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.671 - 2.06i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (1.15 + 0.841i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (4.94 - 3.59i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-0.545 - 1.67i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (0.696 - 2.14i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-1.83 + 1.33i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.70 + 5.23i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.722 - 2.22i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (2.25 - 1.63i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (2.41 - 7.44i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.785425952471902725830641375406, −8.295513663907513467524980799385, −7.56703012652629223509899569402, −7.34313422341686415096472780741, −6.12264293554298314568929913644, −5.05544781252197751708182101273, −3.98327489612706871849548166033, −2.75734297380717353674561387536, −2.09485176462322183406265951335, −0.28520338306779711560166187589, 2.73106778421303302523702854786, 3.50527840108314593044309376843, 4.50689257733933472036198622428, 4.90870018867171317918730862958, 6.12330883088195358722914729494, 7.27551405141862542124483080837, 8.189553212607673111039342755536, 8.705663291918266585045746699991, 9.707698840474211822168893657344, 10.64572304888526963686098481064

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