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Properties

Label 2-950-19.16-c1-0-1
Degree $2$
Conductor $950$
Sign $-0.903 + 0.427i$
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  + (0.766 + 0.642i)2-s + (−2.22 + 0.811i)3-s + (0.173 + 0.984i)4-s + (−2.22 − 0.811i)6-s + (1.57 + 2.73i)7-s + (−0.500 + 0.866i)8-s + (2.00 − 1.68i)9-s + (−0.688 + 1.19i)11-s + (−1.18 − 2.05i)12-s + (−4.06 − 1.47i)13-s + (−0.547 + 3.10i)14-s + (−0.939 + 0.342i)16-s + (0.993 + 0.833i)17-s + 2.62·18-s + (1.08 + 4.22i)19-s + ⋯
L(s)  = 1  + (0.541 + 0.454i)2-s + (−1.28 + 0.468i)3-s + (0.0868 + 0.492i)4-s + (−0.909 − 0.331i)6-s + (0.596 + 1.03i)7-s + (−0.176 + 0.306i)8-s + (0.669 − 0.562i)9-s + (−0.207 + 0.359i)11-s + (−0.342 − 0.592i)12-s + (−1.12 − 0.410i)13-s + (−0.146 + 0.830i)14-s + (−0.234 + 0.0855i)16-s + (0.240 + 0.202i)17-s + 0.618·18-s + (0.249 + 0.968i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.903 + 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.903 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.903 + 0.427i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.903 + 0.427i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.163172 - 0.725920i\)
\(L(\frac12)\) \(\approx\) \(0.163172 - 0.725920i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 - 0.642i)T \)
5 \( 1 \)
19 \( 1 + (-1.08 - 4.22i)T \)
good3 \( 1 + (2.22 - 0.811i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (-1.57 - 2.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.688 - 1.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.06 + 1.47i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-0.993 - 0.833i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-0.369 - 2.09i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (0.0998 - 0.0837i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.173 + 0.300i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 10.3T + 37T^{2} \)
41 \( 1 + (10.3 - 3.76i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-2.00 + 11.3i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (0.0585 - 0.0491i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (1.08 + 6.12i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (6.27 + 5.26i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.36 - 7.72i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-0.789 + 0.662i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-2.02 + 11.4i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (11.5 - 4.18i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (13.1 - 4.79i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-5.68 - 9.85i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-17.1 - 6.23i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-12.4 - 10.4i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.50490006587843845387649129759, −9.940487511307999307000711610566, −8.758025234573588616732739306585, −7.86328653999609086669099859647, −6.93603436825855727147363597928, −5.88776535157773938137371304617, −5.23737811164745607055230296630, −4.88414025020167141465265441541, −3.52461811685514964837626111539, −2.05788925136383035530658833708, 0.33666513298142618957609648896, 1.58541103724665311535431600907, 3.09899465696594958305302378927, 4.56969741455908236837414897070, 4.97715811176199888818725593308, 5.99448798357148538178824304870, 6.99087235518718115764152363804, 7.42369201361618020121545503027, 8.802401674383783804689664165272, 10.00498507367966441282662334614

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