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Properties

Label 2-950-95.44-c1-0-29
Degree $2$
Conductor $950$
Sign $-0.779 + 0.626i$
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  + (0.642 + 0.766i)2-s + (0.936 − 2.57i)3-s + (−0.173 + 0.984i)4-s + (2.57 − 0.936i)6-s + (−3.33 − 1.92i)7-s + (−0.866 + 0.500i)8-s + (−3.44 − 2.89i)9-s + (−2.86 − 4.96i)11-s + (2.37 + 1.36i)12-s + (1.84 + 5.05i)13-s + (−0.668 − 3.79i)14-s + (−0.939 − 0.342i)16-s + (0.883 + 1.05i)17-s − 4.50i·18-s + (−4.23 − 1.03i)19-s + ⋯
L(s)  = 1  + (0.454 + 0.541i)2-s + (0.540 − 1.48i)3-s + (−0.0868 + 0.492i)4-s + (1.05 − 0.382i)6-s + (−1.25 − 0.727i)7-s + (−0.306 + 0.176i)8-s + (−1.14 − 0.964i)9-s + (−0.863 − 1.49i)11-s + (0.684 + 0.395i)12-s + (0.510 + 1.40i)13-s + (−0.178 − 1.01i)14-s + (−0.234 − 0.0855i)16-s + (0.214 + 0.255i)17-s − 1.06i·18-s + (−0.971 − 0.238i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.779 + 0.626i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.779 + 0.626i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.418587 - 1.18872i\)
\(L(\frac12)\) \(\approx\) \(0.418587 - 1.18872i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.642 - 0.766i)T \)
5 \( 1 \)
19 \( 1 + (4.23 + 1.03i)T \)
good3 \( 1 + (-0.936 + 2.57i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (3.33 + 1.92i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.86 + 4.96i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.84 - 5.05i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-0.883 - 1.05i)T + (-2.95 + 16.7i)T^{2} \)
23 \( 1 + (1.55 + 0.274i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (5.22 + 4.38i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-3.07 + 5.32i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.89iT - 37T^{2} \)
41 \( 1 + (3.39 + 1.23i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-3.22 + 0.568i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (4.65 - 5.55i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (-11.5 - 2.04i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-2.00 + 1.68i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (1.05 - 5.99i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-4.03 + 4.81i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (2.16 + 12.2i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-3.22 + 8.85i)T + (-55.9 - 46.9i)T^{2} \)
79 \( 1 + (-7.27 - 2.64i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (3.88 + 2.24i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.964 - 0.351i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (0.849 + 1.01i)T + (-16.8 + 95.5i)T^{2} \)
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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

−9.407182335686264946785208571363, −8.567404124464517489264878526954, −7.88237019468067397324642720081, −7.07761862218326048676438071625, −6.30710861959109115942754060462, −5.94029653771339288907153359447, −4.14259668907128320937865555101, −3.27369376154140029764274856712, −2.19113084459853657013899527967, −0.43801040527087364873162541459, 2.38149561935312464393117718166, 3.16466076159827701113964801834, 3.92522289959547851322101858614, 5.04106836966793980004004533398, 5.59709228996625789309884325908, 6.84240233437925977685996005626, 8.203916526815608624378896019447, 8.981740654746125025238908355015, 9.947148494643951908527792925151, 10.09832106230461784891736178345

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