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Properties

Label 2-950-25.16-c1-0-40
Degree $2$
Conductor $950$
Sign $0.996 + 0.0801i$
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  + (0.809 + 0.587i)2-s + (0.188 + 0.579i)3-s + (0.309 + 0.951i)4-s + (1.27 − 1.83i)5-s + (−0.188 + 0.579i)6-s + 0.873·7-s + (−0.309 + 0.951i)8-s + (2.12 − 1.54i)9-s + (2.11 − 0.739i)10-s + (−4.34 − 3.15i)11-s + (−0.493 + 0.358i)12-s + (3.53 − 2.57i)13-s + (0.706 + 0.513i)14-s + (1.30 + 0.391i)15-s + (−0.809 + 0.587i)16-s + (1.22 − 3.75i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.108 + 0.334i)3-s + (0.154 + 0.475i)4-s + (0.569 − 0.822i)5-s + (−0.0769 + 0.236i)6-s + 0.330·7-s + (−0.109 + 0.336i)8-s + (0.708 − 0.514i)9-s + (0.667 − 0.233i)10-s + (−1.31 − 0.951i)11-s + (−0.142 + 0.103i)12-s + (0.981 − 0.713i)13-s + (0.188 + 0.137i)14-s + (0.337 + 0.101i)15-s + (−0.202 + 0.146i)16-s + (0.296 − 0.911i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.996 + 0.0801i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.996 + 0.0801i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.59610 - 0.104192i\)
\(L(\frac12)\) \(\approx\) \(2.59610 - 0.104192i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (-1.27 + 1.83i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
good3 \( 1 + (-0.188 - 0.579i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 - 0.873T + 7T^{2} \)
11 \( 1 + (4.34 + 3.15i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-3.53 + 2.57i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.22 + 3.75i)T + (-13.7 - 9.99i)T^{2} \)
23 \( 1 + (0.985 + 0.716i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-2.18 - 6.73i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.90 - 8.95i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.47 - 3.24i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-8.26 + 6.00i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 3.84T + 43T^{2} \)
47 \( 1 + (0.634 + 1.95i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.58 - 4.88i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-2.99 + 2.17i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-11.2 - 8.19i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (1.78 - 5.49i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-3.19 - 9.82i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.17 - 1.58i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.37 + 4.22i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.371 - 1.14i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (5.87 + 4.27i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-5.09 - 15.6i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.15554195454784002956535273943, −8.890093276353297954714864317798, −8.519227118208731683625864052506, −7.49058175074533033020916622369, −6.46782722801916028360227499879, −5.29251199811270343974812285282, −5.15122352453988545238017982269, −3.78877528410383403210935111933, −2.84067536062772213992513558912, −1.09984805349872496045988582701, 1.77761372487942731604367195308, 2.32066968016352399908404993994, 3.72656218718457698668034356973, 4.67597306349505745665461079534, 5.75226931897904572694327840856, 6.50882613607790935666696369877, 7.51679908990787750291789374319, 8.121943657888965597997710224809, 9.645306389560246184879622862813, 10.11670174562675043824704192363

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