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Properties

Label 2-950-19.6-c1-0-1
Degree $2$
Conductor $950$
Sign $-0.702 + 0.711i$
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  + (−0.766 + 0.642i)2-s + (0.799 + 0.291i)3-s + (0.173 − 0.984i)4-s + (−0.799 + 0.291i)6-s + (−2.52 + 4.37i)7-s + (0.500 + 0.866i)8-s + (−1.74 − 1.46i)9-s + (1.10 + 1.92i)11-s + (0.425 − 0.737i)12-s + (−4.57 + 1.66i)13-s + (−0.876 − 4.97i)14-s + (−0.939 − 0.342i)16-s + (3.35 − 2.81i)17-s + 2.27·18-s + (−3.62 − 2.42i)19-s + ⋯
L(s)  = 1  + (−0.541 + 0.454i)2-s + (0.461 + 0.168i)3-s + (0.0868 − 0.492i)4-s + (−0.326 + 0.118i)6-s + (−0.954 + 1.65i)7-s + (0.176 + 0.306i)8-s + (−0.580 − 0.487i)9-s + (0.334 + 0.579i)11-s + (0.122 − 0.212i)12-s + (−1.26 + 0.462i)13-s + (−0.234 − 1.32i)14-s + (−0.234 − 0.0855i)16-s + (0.812 − 0.682i)17-s + 0.536·18-s + (−0.831 − 0.556i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.702 + 0.711i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.702 + 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0513147 - 0.122827i\)
\(L(\frac12)\) \(\approx\) \(0.0513147 - 0.122827i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 - 0.642i)T \)
5 \( 1 \)
19 \( 1 + (3.62 + 2.42i)T \)
good3 \( 1 + (-0.799 - 0.291i)T + (2.29 + 1.92i)T^{2} \)
7 \( 1 + (2.52 - 4.37i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.10 - 1.92i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.57 - 1.66i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-3.35 + 2.81i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-0.792 + 4.49i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (1.48 + 1.24i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-3.76 + 6.51i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.47T + 37T^{2} \)
41 \( 1 + (9.83 + 3.58i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.46 - 8.29i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (3.69 + 3.10i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (1.77 - 10.0i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (3.98 - 3.34i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.909 - 5.15i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-1.13 - 0.949i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-2.23 - 12.6i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (12.4 + 4.54i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-5.71 - 2.08i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (6.23 - 10.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.85 - 2.13i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (1.50 - 1.26i)T + (16.8 - 95.5i)T^{2} \)
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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

−9.988626500941117287202218848542, −9.594319283744984313054095310484, −8.954051983851923720544589306186, −8.326612818322362574415899659636, −7.13898216771542026855459400587, −6.37710456722453464397248596989, −5.59031581138561503762835669566, −4.51166315785063215658563026251, −2.92735517711237612826338356896, −2.31428018528058184480511544381, 0.06586590762222884060046327270, 1.61910947886246334519498946672, 3.17801501328028705241699413067, 3.58914695807714898651886255512, 4.99252804255988745654165251467, 6.32698071301836286861827198435, 7.23935054421705927140914826737, 7.88184590644232604000843162282, 8.653837204849756613498771213350, 9.740935897140541888291311623222

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