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Properties

Label 2-950-95.8-c1-0-17
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: $\chi_{950} (293, \cdot )$
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

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

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