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Properties

Label 2-950-95.54-c1-0-9
Degree $2$
Conductor $950$
Sign $-0.362 - 0.931i$
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 + (0.934 + 2.56i)3-s + (−0.173 − 0.984i)4-s + (2.56 + 0.934i)6-s + (−1.85 + 1.06i)7-s + (−0.866 − 0.500i)8-s + (−3.42 + 2.87i)9-s + (0.926 − 1.60i)11-s + (2.36 − 1.36i)12-s + (−1.88 + 5.18i)13-s + (−0.371 + 2.10i)14-s + (−0.939 + 0.342i)16-s + (−2.37 + 2.82i)17-s + 4.46i·18-s + (−1.21 + 4.18i)19-s + ⋯
L(s)  = 1  + (0.454 − 0.541i)2-s + (0.539 + 1.48i)3-s + (−0.0868 − 0.492i)4-s + (1.04 + 0.381i)6-s + (−0.699 + 0.403i)7-s + (−0.306 − 0.176i)8-s + (−1.14 + 0.957i)9-s + (0.279 − 0.483i)11-s + (0.683 − 0.394i)12-s + (−0.522 + 1.43i)13-s + (−0.0991 + 0.562i)14-s + (−0.234 + 0.0855i)16-s + (−0.575 + 0.686i)17-s + 1.05i·18-s + (−0.278 + 0.960i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.362 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.362 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.362 - 0.931i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.362 - 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.971211 + 1.42017i\)
\(L(\frac12)\) \(\approx\) \(0.971211 + 1.42017i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.21 - 4.18i)T \)
good3 \( 1 + (-0.934 - 2.56i)T + (-2.29 + 1.92i)T^{2} \)
7 \( 1 + (1.85 - 1.06i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.926 + 1.60i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.88 - 5.18i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (2.37 - 2.82i)T + (-2.95 - 16.7i)T^{2} \)
23 \( 1 + (2.89 - 0.511i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (-5.36 + 4.50i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-2.12 - 3.68i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.49iT - 37T^{2} \)
41 \( 1 + (-1.47 + 0.536i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (2.29 + 0.404i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (-7.20 - 8.59i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (-6.32 + 1.11i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (10.8 + 9.07i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (0.0461 + 0.261i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-9.49 - 11.3i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-0.722 + 4.09i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (5.27 + 14.4i)T + (-55.9 + 46.9i)T^{2} \)
79 \( 1 + (2.70 - 0.985i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-12.1 + 7.00i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.86 - 0.676i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (6.94 - 8.28i)T + (-16.8 - 95.5i)T^{2} \)
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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

−10.26945318769970043985964570739, −9.488497229357702154122055144083, −9.054547976410130975581128285510, −8.158321068770681007728318328738, −6.57078835659867269320422585832, −5.84403097410973362556723483729, −4.57779256739258936612455394171, −4.06805158913051666348287231736, −3.18058463947032838160550420255, −2.10751067359923700647118661377, 0.61945406543807949707862984445, 2.38730081124160110777665923177, 3.13366572194106891833276454943, 4.50763253779997079241394532571, 5.66174918238513137877984152374, 6.79625163084550565682721785891, 6.97319169251681462208635585313, 7.908535941627121953889329938130, 8.594582834296995256502462256349, 9.607285251623875659724935757894

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