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Properties

Label 2-950-19.17-c1-0-8
Degree $2$
Conductor $950$
Sign $0.434 - 0.900i$
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  + (0.173 − 0.984i)2-s + (0.849 + 0.712i)3-s + (−0.939 − 0.342i)4-s + (0.849 − 0.712i)6-s + (2.46 + 4.26i)7-s + (−0.5 + 0.866i)8-s + (−0.307 − 1.74i)9-s + (−2.20 + 3.82i)11-s + (−0.554 − 0.960i)12-s + (−2.02 + 1.69i)13-s + (4.63 − 1.68i)14-s + (0.766 + 0.642i)16-s + (−0.872 + 4.94i)17-s − 1.76·18-s + (−4.21 − 1.11i)19-s + ⋯
L(s)  = 1  + (0.122 − 0.696i)2-s + (0.490 + 0.411i)3-s + (−0.469 − 0.171i)4-s + (0.346 − 0.291i)6-s + (0.931 + 1.61i)7-s + (−0.176 + 0.306i)8-s + (−0.102 − 0.580i)9-s + (−0.665 + 1.15i)11-s + (−0.160 − 0.277i)12-s + (−0.560 + 0.470i)13-s + (1.23 − 0.450i)14-s + (0.191 + 0.160i)16-s + (−0.211 + 1.20i)17-s − 0.417·18-s + (−0.966 − 0.256i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.434 - 0.900i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.434 - 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36651 + 0.858080i\)
\(L(\frac12)\) \(\approx\) \(1.36651 + 0.858080i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 + 0.984i)T \)
5 \( 1 \)
19 \( 1 + (4.21 + 1.11i)T \)
good3 \( 1 + (-0.849 - 0.712i)T + (0.520 + 2.95i)T^{2} \)
7 \( 1 + (-2.46 - 4.26i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.20 - 3.82i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.02 - 1.69i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (0.872 - 4.94i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (0.964 + 0.351i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.462 - 2.62i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (1.01 + 1.76i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.00T + 37T^{2} \)
41 \( 1 + (-1.65 - 1.38i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-1.41 + 0.516i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (0.310 + 1.76i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (-5.28 - 1.92i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-2.44 + 13.8i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-13.6 - 4.96i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-2.36 - 13.4i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-14.6 + 5.33i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-9.66 - 8.10i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (2.17 + 1.82i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (3.64 + 6.30i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.72 + 6.48i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (2.06 - 11.7i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.09772977118292482773950161404, −9.393296999278129700602632548852, −8.630734990495260307869407105100, −8.135968763389458395120651892232, −6.71677317172025057974516564268, −5.58331839971828829283439119270, −4.76127933625945007078334826956, −3.91117869446082893422507669690, −2.47120208117334692366285034056, −2.00945082518316490524516848109, 0.66448382961947392519418023869, 2.36625934736671080551480152227, 3.65845062431829695893641487944, 4.73977151363421705816455967759, 5.42661996727647281027192303508, 6.76659980451119806903672751042, 7.50424177158248969635353044394, 8.029359507507118816740013666076, 8.612384807307165067176644348164, 9.920288705844502913303159579580

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