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Properties

Label 2-950-25.16-c1-0-43
Degree $2$
Conductor $950$
Sign $0.974 - 0.225i$
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 + (−1.01 − 3.11i)3-s + (0.309 + 0.951i)4-s + (−2.01 − 0.975i)5-s + (−1.01 + 3.11i)6-s − 3.17·7-s + (0.309 − 0.951i)8-s + (−6.27 + 4.55i)9-s + (1.05 + 1.97i)10-s + (−3.43 − 2.49i)11-s + (2.65 − 1.92i)12-s + (−0.742 + 0.539i)13-s + (2.56 + 1.86i)14-s + (−1.00 + 7.26i)15-s + (−0.809 + 0.587i)16-s + (2.27 − 7.01i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (−0.584 − 1.80i)3-s + (0.154 + 0.475i)4-s + (−0.899 − 0.436i)5-s + (−0.413 + 1.27i)6-s − 1.19·7-s + (0.109 − 0.336i)8-s + (−2.09 + 1.51i)9-s + (0.333 + 0.623i)10-s + (−1.03 − 0.751i)11-s + (0.765 − 0.556i)12-s + (−0.205 + 0.149i)13-s + (0.686 + 0.498i)14-s + (−0.259 + 1.87i)15-s + (−0.202 + 0.146i)16-s + (0.552 − 1.70i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.974 - 0.225i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.974 - 0.225i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0929529 + 0.0106301i\)
\(L(\frac12)\) \(\approx\) \(0.0929529 + 0.0106301i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.01 + 0.975i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
good3 \( 1 + (1.01 + 3.11i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + 3.17T + 7T^{2} \)
11 \( 1 + (3.43 + 2.49i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.742 - 0.539i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.27 + 7.01i)T + (-13.7 - 9.99i)T^{2} \)
23 \( 1 + (5.17 + 3.76i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.29 - 3.99i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.0610 + 0.187i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-9.45 + 6.87i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (0.353 - 0.257i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 4.77T + 43T^{2} \)
47 \( 1 + (0.948 + 2.91i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.96 + 6.05i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-2.18 + 1.58i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.438 + 0.318i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (3.37 - 10.3i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (3.74 + 11.5i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-3.06 - 2.22i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (4.09 + 12.6i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.90 - 12.0i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-3.41 - 2.48i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-1.84 - 5.67i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.056421170507933558305970673918, −8.181064207684870306376381395113, −7.56848614445761231483816611311, −6.93474068523110336882116985223, −6.02160866441683699775786369748, −5.00578915587068610254344962246, −3.23863178201887706528100892762, −2.44843495179085480185303487057, −0.72966796866631816009243975230, −0.088872732436515746844860749101, 2.93217332542612732931376051163, 3.82437417619895154649058043575, 4.64310048950556046899242623348, 5.81723687975080805813816611847, 6.34175984493760507067602387359, 7.68551057112336011214844693387, 8.357820744590716403569052725885, 9.541052869612753801943555051976, 10.15260516135118018143554754450, 10.32319698017409830515483792481

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