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Properties

Label 2-950-95.27-c1-0-2
Degree $2$
Conductor $950$
Sign $-0.408 - 0.912i$
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  + (−0.258 − 0.965i)2-s + (1.26 − 0.338i)3-s + (−0.866 + 0.499i)4-s + (−0.653 − 1.13i)6-s + (−1.48 − 1.48i)7-s + (0.707 + 0.707i)8-s + (−1.11 + 0.646i)9-s − 6.07·11-s + (−0.923 + 0.923i)12-s + (−0.411 + 1.53i)13-s + (−1.05 + 1.82i)14-s + (0.500 − 0.866i)16-s + (2.57 − 0.689i)17-s + (0.913 + 0.913i)18-s + (0.396 + 4.34i)19-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (0.728 − 0.195i)3-s + (−0.433 + 0.249i)4-s + (−0.266 − 0.461i)6-s + (−0.562 − 0.562i)7-s + (0.249 + 0.249i)8-s + (−0.373 + 0.215i)9-s − 1.83·11-s + (−0.266 + 0.266i)12-s + (−0.114 + 0.426i)13-s + (−0.281 + 0.486i)14-s + (0.125 − 0.216i)16-s + (0.624 − 0.167i)17-s + (0.215 + 0.215i)18-s + (0.0908 + 0.995i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.408 - 0.912i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.408 - 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0751886 + 0.116015i\)
\(L(\frac12)\) \(\approx\) \(0.0751886 + 0.116015i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 \)
19 \( 1 + (-0.396 - 4.34i)T \)
good3 \( 1 + (-1.26 + 0.338i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (1.48 + 1.48i)T + 7iT^{2} \)
11 \( 1 + 6.07T + 11T^{2} \)
13 \( 1 + (0.411 - 1.53i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (-2.57 + 0.689i)T + (14.7 - 8.5i)T^{2} \)
23 \( 1 + (7.49 + 2.00i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.31 - 2.28i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.53iT - 31T^{2} \)
37 \( 1 + (-5.03 + 5.03i)T - 37iT^{2} \)
41 \( 1 + (5.76 + 3.33i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.566 + 2.11i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (0.135 - 0.505i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.625 - 2.33i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (3.20 - 5.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.49 + 12.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.0 + 2.95i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3.66 - 2.11i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.859 + 3.20i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.32 - 5.75i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.732 - 0.732i)T - 83iT^{2} \)
89 \( 1 + (-0.347 - 0.602i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.272 - 1.01i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.34664275157961622062625878533, −9.634114744216082944038041698928, −8.564745248718817881136501109006, −7.951337787981998836342951853713, −7.29832616540001183176440454615, −5.90660015186929010271270345957, −4.93567065158571770100131107553, −3.65269809643981397968942714970, −2.89056731531841544205528022134, −1.89125164132098575391457838173, 0.05697719643914448881734937976, 2.46822723087646569641477397187, 3.19063342966284463895107775912, 4.55353072629892543178699034405, 5.66803397598352692836755232651, 6.15281939854665735055241231078, 7.64381667877428335087338185705, 7.980265086032746402265343044136, 8.815509259316326279674594636722, 9.780457792213280688568692727738

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