Properties

Label 2-950-95.12-c1-0-17
Degree $2$
Conductor $950$
Sign $0.876 - 0.481i$
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  + (0.965 + 0.258i)2-s + (0.258 − 0.965i)3-s + (0.866 + 0.499i)4-s + (0.499 − 0.866i)6-s + (2.44 + 2.44i)7-s + (0.707 + 0.707i)8-s + (1.73 + i)9-s + (0.707 − 0.707i)12-s + (−1.93 + 0.517i)13-s + (1.73 + 2.99i)14-s + (0.500 + 0.866i)16-s + (−0.448 + 1.67i)17-s + (1.41 + 1.41i)18-s + (−2.59 + 3.5i)19-s + (2.99 − 1.73i)21-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.149 − 0.557i)3-s + (0.433 + 0.249i)4-s + (0.204 − 0.353i)6-s + (0.925 + 0.925i)7-s + (0.249 + 0.249i)8-s + (0.577 + 0.333i)9-s + (0.204 − 0.204i)12-s + (−0.535 + 0.143i)13-s + (0.462 + 0.801i)14-s + (0.125 + 0.216i)16-s + (−0.108 + 0.405i)17-s + (0.333 + 0.333i)18-s + (−0.596 + 0.802i)19-s + (0.654 − 0.377i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.876 - 0.481i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.876 - 0.481i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.73348 + 0.700655i\)
\(L(\frac12)\) \(\approx\) \(2.73348 + 0.700655i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 \)
19 \( 1 + (2.59 - 3.5i)T \)
good3 \( 1 + (-0.258 + 0.965i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-2.44 - 2.44i)T + 7iT^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (1.93 - 0.517i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.448 - 1.67i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (-0.896 - 3.34i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.73 + 3i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 10.3iT - 31T^{2} \)
37 \( 1 + (2.82 - 2.82i)T - 37iT^{2} \)
41 \( 1 + (-6 + 3.46i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.34 - 0.896i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (6.69 - 1.79i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.59 - 4.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.07 + 7.72i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-12 + 6.92i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (8.36 + 2.24i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.22 + 1.22i)T - 83iT^{2} \)
89 \( 1 + (-4.33 + 7.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.76 + 1.81i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.18006853025723505175027620755, −9.166842003581906485345721146149, −7.997875213933194328712722787191, −7.78199407158578407211693019565, −6.59418141375441604899743473604, −5.76944444476687714527193307769, −4.86125101467580047395786828938, −3.99357518365721313163860900712, −2.45123182907009929689869705329, −1.74080078547885216288148132063, 1.19198183968524442303711609186, 2.67970205679519785361093339440, 3.87006814168156895008408541210, 4.62562232559192454072165385449, 5.17468493974355561078866005380, 6.73489369051495569831058556304, 7.17094681208925797420342078011, 8.326994942536348145012605940690, 9.251326743010273927828092054532, 10.27975704009064939652616758992

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