Properties

Label 2-950-95.4-c1-0-2
Degree $2$
Conductor $950$
Sign $-0.797 + 0.602i$
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  + (−0.342 + 0.939i)2-s + (−3.16 + 0.558i)3-s + (−0.766 − 0.642i)4-s + (0.558 − 3.16i)6-s + (0.0202 + 0.0116i)7-s + (0.866 − 0.500i)8-s + (6.89 − 2.51i)9-s + (−1.08 − 1.87i)11-s + (2.78 + 1.60i)12-s + (1.56 + 0.276i)13-s + (−0.0179 + 0.0150i)14-s + (0.173 + 0.984i)16-s + (−2.72 + 7.49i)17-s + 7.34i·18-s + (−1.06 + 4.22i)19-s + ⋯
L(s)  = 1  + (−0.241 + 0.664i)2-s + (−1.82 + 0.322i)3-s + (−0.383 − 0.321i)4-s + (0.227 − 1.29i)6-s + (0.00765 + 0.00442i)7-s + (0.306 − 0.176i)8-s + (2.29 − 0.836i)9-s + (−0.325 − 0.564i)11-s + (0.803 + 0.464i)12-s + (0.434 + 0.0765i)13-s + (−0.00478 + 0.00401i)14-s + (0.0434 + 0.246i)16-s + (−0.661 + 1.81i)17-s + 1.73i·18-s + (−0.244 + 0.969i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.797 + 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.797 + 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.797 + 0.602i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.797 + 0.602i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0486956 - 0.145259i\)
\(L(\frac12)\) \(\approx\) \(0.0486956 - 0.145259i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.342 - 0.939i)T \)
5 \( 1 \)
19 \( 1 + (1.06 - 4.22i)T \)
good3 \( 1 + (3.16 - 0.558i)T + (2.81 - 1.02i)T^{2} \)
7 \( 1 + (-0.0202 - 0.0116i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.08 + 1.87i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.56 - 0.276i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (2.72 - 7.49i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-2.88 + 3.43i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-7.09 + 2.58i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (2.22 - 3.86i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.389iT - 37T^{2} \)
41 \( 1 + (-0.972 - 5.51i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (6.23 + 7.43i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (1.46 + 4.01i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (2.43 - 2.89i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (3.41 + 1.24i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (8.84 + 7.41i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (1.31 + 3.62i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (9.85 - 8.27i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (1.87 - 0.330i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (-1.65 - 9.36i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (2.27 + 1.31i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.54 - 8.74i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-3.81 + 10.4i)T + (-74.3 - 62.3i)T^{2} \)
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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

−10.54931907239025543996110706389, −10.04524157950895248823384955337, −8.758573270109904629496742506263, −8.038490990416737634645261812606, −6.64857836718884781365913081361, −6.34747863943350638676533144011, −5.54092082379097215017048410318, −4.68395635837570082441949439559, −3.77965756284315473668404480481, −1.40943405930540892796941132539, 0.11369518784451763026958731151, 1.35534081050781733785797506917, 2.82478331009545275045079069329, 4.61223297646086349781547501004, 4.89779481063626762735162585891, 6.06008652717669708315902344469, 6.98548773048003334655059818564, 7.55941333387788302176862391577, 9.001746030903912147172364107653, 9.779537749964202351882236173705

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