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Properties

Label 2-950-95.12-c1-0-1
Degree $2$
Conductor $950$
Sign $-0.0429 - 0.999i$
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  + (−0.965 − 0.258i)2-s + (0.338 − 1.26i)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 + (−1.53 + 0.411i)13-s + (1.05 + 1.82i)14-s + (0.500 + 0.866i)16-s + (−0.689 + 2.57i)17-s + (−0.913 − 0.913i)18-s + (−0.396 + 4.34i)19-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.195 − 0.728i)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 + (−0.426 + 0.114i)13-s + (0.281 + 0.486i)14-s + (0.125 + 0.216i)16-s + (−0.167 + 0.624i)17-s + (−0.215 − 0.215i)18-s + (−0.0908 + 0.995i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0429 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0429 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.0429 - 0.999i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.0429 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.250616 + 0.261610i\)
\(L(\frac12)\) \(\approx\) \(0.250616 + 0.261610i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 \)
19 \( 1 + (0.396 - 4.34i)T \)
good3 \( 1 + (-0.338 + 1.26i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (1.48 + 1.48i)T + 7iT^{2} \)
11 \( 1 + 6.07T + 11T^{2} \)
13 \( 1 + (1.53 - 0.411i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.689 - 2.57i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (-2.00 - 7.49i)T + (-19.9 + 11.5i)T^{2} \)
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 \( 1 + (-2.11 - 0.566i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-0.505 + 0.135i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.33 - 0.625i)T + (45.8 - 26.5i)T^{2} \)
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 \( 1 + (2.95 + 11.0i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3.66 + 2.11i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.20 - 0.859i)T + (63.2 + 36.5i)T^{2} \)
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 \( 1 + (-1.01 - 0.272i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.21936134789475935004952506230, −9.605805309750668870655496526660, −8.359388958453837632437924501130, −7.67825348347866502247201867882, −7.26787644648745893928617113242, −6.19631526919811025056849308276, −5.11052683231024277824421665369, −3.71275827249406237723500913052, −2.58007873145956948551138533294, −1.50882676144011321365419189805, 0.19752890073094571321920106774, 2.40724786215150846399287638055, 3.14897340416361294205889794444, 4.73353808354637075795665296631, 5.32197116687882170427493789861, 6.62771578807653310096869570332, 7.30333457676175849177615612260, 8.398263581682211783445304655382, 9.046123768360651128622986388736, 9.787727982720119327736023550921

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