Properties

Label 2-950-19.7-c1-0-5
Degree $2$
Conductor $950$
Sign $0.444 - 0.895i$
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  + (0.5 + 0.866i)2-s + (−1.51 − 2.62i)3-s + (−0.499 + 0.866i)4-s + (1.51 − 2.62i)6-s − 4.93·7-s − 0.999·8-s + (−3.10 + 5.38i)9-s + 1.28·11-s + 3.03·12-s + (1.98 − 3.43i)13-s + (−2.46 − 4.27i)14-s + (−0.5 − 0.866i)16-s + (1.10 + 1.91i)17-s − 6.21·18-s + (3.12 + 3.03i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.876 − 1.51i)3-s + (−0.249 + 0.433i)4-s + (0.619 − 1.07i)6-s − 1.86·7-s − 0.353·8-s + (−1.03 + 1.79i)9-s + 0.386·11-s + 0.876·12-s + (0.550 − 0.952i)13-s + (−0.659 − 1.14i)14-s + (−0.125 − 0.216i)16-s + (0.268 + 0.465i)17-s − 1.46·18-s + (0.716 + 0.697i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.444 - 0.895i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.444 - 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.613997 + 0.380687i\)
\(L(\frac12)\) \(\approx\) \(0.613997 + 0.380687i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 \)
19 \( 1 + (-3.12 - 3.03i)T \)
good3 \( 1 + (1.51 + 2.62i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 4.93T + 7T^{2} \)
11 \( 1 - 1.28T + 11T^{2} \)
13 \( 1 + (-1.98 + 3.43i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.10 - 1.91i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (3.84 - 6.65i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.14 + 3.70i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.14T + 31T^{2} \)
37 \( 1 - 9.38T + 37T^{2} \)
41 \( 1 + (-2.30 - 3.98i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.32 - 5.75i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.46 - 6.00i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.46 + 6.00i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.15 - 5.47i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.64 - 9.77i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.965 + 1.67i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.16 + 7.20i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.41 - 5.91i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.76 + 3.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.28T + 83T^{2} \)
89 \( 1 + (4.46 - 7.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.07 - 1.85i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.12241506741515927455642635862, −9.352687085585181953294972860264, −7.990020188325863699582310659708, −7.54017274934103360782905931486, −6.56405254957959445849696453653, −5.94828922139835310944776846776, −5.67075155095553773009518818478, −3.86017231633254803269085929043, −2.84730480058810184550735860944, −1.08513734195960753942756619999, 0.40388572879925667267746415683, 2.81298447741939950013799980797, 3.74188997990561928370729707229, 4.32215986321796936502901510687, 5.43124371878225861698634978703, 6.22677862826822547780292238577, 6.87305993243524371119572568183, 8.913768851488844278804828753798, 9.363351056639092321968482429976, 9.973399656844925684662854930353

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