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Properties

Label 2-950-19.11-c1-0-23
Degree $2$
Conductor $950$
Sign $0.221 + 0.975i$
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  + (0.5 − 0.866i)2-s + (0.664 − 1.15i)3-s + (−0.499 − 0.866i)4-s + (−0.664 − 1.15i)6-s + 2.32·7-s − 0.999·8-s + (0.616 + 1.06i)9-s + 6.39·11-s − 1.32·12-s + (0.429 + 0.743i)13-s + (1.16 − 2.01i)14-s + (−0.5 + 0.866i)16-s + (−2.34 + 4.06i)17-s + 1.23·18-s + (3.75 − 2.21i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.383 − 0.664i)3-s + (−0.249 − 0.433i)4-s + (−0.271 − 0.469i)6-s + 0.880·7-s − 0.353·8-s + (0.205 + 0.355i)9-s + 1.92·11-s − 0.383·12-s + (0.119 + 0.206i)13-s + (0.311 − 0.539i)14-s + (−0.125 + 0.216i)16-s + (−0.568 + 0.985i)17-s + 0.290·18-s + (0.860 − 0.509i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.221 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.221 + 0.975i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.221 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01216 - 1.60634i\)
\(L(\frac12)\) \(\approx\) \(2.01216 - 1.60634i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 \)
19 \( 1 + (-3.75 + 2.21i)T \)
good3 \( 1 + (-0.664 + 1.15i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 2.32T + 7T^{2} \)
11 \( 1 - 6.39T + 11T^{2} \)
13 \( 1 + (-0.429 - 0.743i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.34 - 4.06i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.73 + 3.00i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.21 - 3.82i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.25T + 31T^{2} \)
37 \( 1 + 9.76T + 37T^{2} \)
41 \( 1 + (1.84 - 3.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.56 + 6.17i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.59 + 6.22i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.10 + 5.37i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.09 + 5.35i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.01 - 6.94i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.45 + 4.24i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.10 - 1.90i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.32 - 4.03i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.79 - 10.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.07T + 83T^{2} \)
89 \( 1 + (5.64 + 9.78i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.67 - 9.82i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.942426319338394921548003957492, −8.816844303690172395916834701507, −8.482513944518071658262909427277, −7.16970606853695673957843703370, −6.61336321418382467141739162368, −5.34216392398065583932203364259, −4.36544867130133885585811870202, −3.50397597573142819098099045350, −1.96595759283898213887532636994, −1.41123166081409293411567953934, 1.47483832602227039304648821405, 3.29061187106906006373359409763, 4.05059036568680158179108688892, 4.81821699422194481113673298727, 5.88526795394336365702027705385, 6.85164382955672137152257095800, 7.60619945614018153056356869345, 8.708769626162057522615074803112, 9.234328811828112258183545514316, 9.912988399368212399420982013094

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