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Properties

Label 2-950-95.12-c1-0-18
Degree $2$
Conductor $950$
Sign $-0.413 - 0.910i$
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 + (−0.633 + 2.36i)3-s + (0.866 + 0.499i)4-s + (−1.22 + 2.12i)6-s + (3.22 + 3.22i)7-s + (0.707 + 0.707i)8-s + (−2.59 − 1.50i)9-s + 3.44·11-s + (−1.73 + 1.73i)12-s + (4.73 − 1.26i)13-s + (2.28 + 3.94i)14-s + (0.500 + 0.866i)16-s + (−0.732 + 2.73i)17-s + (−2.12 − 2.12i)18-s + (−1.25 − 4.17i)19-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.366 + 1.36i)3-s + (0.433 + 0.249i)4-s + (−0.499 + 0.866i)6-s + (1.21 + 1.21i)7-s + (0.249 + 0.249i)8-s + (−0.866 − 0.500i)9-s + 1.04·11-s + (−0.499 + 0.500i)12-s + (1.31 − 0.351i)13-s + (0.609 + 1.05i)14-s + (0.125 + 0.216i)16-s + (−0.177 + 0.662i)17-s + (−0.500 − 0.500i)18-s + (−0.287 − 0.957i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.413 - 0.910i$
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: \((2,\ 950,\ (\ :1/2),\ -0.413 - 0.910i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42343 + 2.20920i\)
\(L(\frac12)\) \(\approx\) \(1.42343 + 2.20920i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.25 + 4.17i)T \)
good3 \( 1 + (0.633 - 2.36i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-3.22 - 3.22i)T + 7iT^{2} \)
11 \( 1 - 3.44T + 11T^{2} \)
13 \( 1 + (-4.73 + 1.26i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.732 - 2.73i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (1.34 + 5.01i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-0.389 + 0.674i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.70iT - 31T^{2} \)
37 \( 1 + (0.389 - 0.389i)T - 37iT^{2} \)
41 \( 1 + (1.5 - 0.866i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.48 + 0.933i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (6.69 - 1.79i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (9.99 - 2.67i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.73 + 3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.47 + 9.22i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (3 - 1.73i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.11 + 0.567i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.97 + 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.34 + 2.34i)T - 83iT^{2} \)
89 \( 1 + (-1.64 + 2.84i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.22 - 2.47i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.56498313293278396884676708910, −9.392417605839536803262648859696, −8.691105600792649003044710588186, −8.038221803562910709480098232301, −6.38802628834091504247736169804, −5.88415377157591071446476464997, −4.84584633883788391759543674806, −4.35590642171420589164773095660, −3.33546077774153091374263485146, −1.88673423616196149523904225165, 1.32402434195231099476874914658, 1.58814582539819620927939755216, 3.53815316450168942919229694890, 4.36661665192980209699328686550, 5.50926965601089382733115247017, 6.50234454072608586153208114434, 7.02517818358026020892814259309, 7.85035136326433061969825305263, 8.653873029516717064682134421488, 10.03662026035814637374886956376

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