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Properties

Label 2-950-95.37-c1-0-11
Degree $2$
Conductor $950$
Sign $-0.0484 + 0.998i$
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  + (−0.707 + 0.707i)2-s + (−1.29 − 1.29i)3-s − 1.00i·4-s + 1.83·6-s + (−2.12 − 2.12i)7-s + (0.707 + 0.707i)8-s + 0.369i·9-s + 5.08·11-s + (−1.29 + 1.29i)12-s + (3.71 + 3.71i)13-s + 3.00·14-s − 1.00·16-s + (1.18 + 1.18i)17-s + (−0.261 − 0.261i)18-s + (−4.18 − 1.20i)19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.749 − 0.749i)3-s − 0.500i·4-s + 0.749·6-s + (−0.803 − 0.803i)7-s + (0.250 + 0.250i)8-s + 0.123i·9-s + 1.53·11-s + (−0.374 + 0.374i)12-s + (1.03 + 1.03i)13-s + 0.803·14-s − 0.250·16-s + (0.287 + 0.287i)17-s + (−0.0616 − 0.0616i)18-s + (−0.960 − 0.276i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0484 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0484 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.0484 + 0.998i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.0484 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.550621 - 0.577961i\)
\(L(\frac12)\) \(\approx\) \(0.550621 - 0.577961i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 \)
19 \( 1 + (4.18 + 1.20i)T \)
good3 \( 1 + (1.29 + 1.29i)T + 3iT^{2} \)
7 \( 1 + (2.12 + 2.12i)T + 7iT^{2} \)
11 \( 1 - 5.08T + 11T^{2} \)
13 \( 1 + (-3.71 - 3.71i)T + 13iT^{2} \)
17 \( 1 + (-1.18 - 1.18i)T + 17iT^{2} \)
23 \( 1 + (-5.59 + 5.59i)T - 23iT^{2} \)
29 \( 1 - 0.861T + 29T^{2} \)
31 \( 1 + 7.04iT - 31T^{2} \)
37 \( 1 + (-3.98 + 3.98i)T - 37iT^{2} \)
41 \( 1 - 2.38iT - 41T^{2} \)
43 \( 1 + (4.14 - 4.14i)T - 43iT^{2} \)
47 \( 1 + (4.04 + 4.04i)T + 47iT^{2} \)
53 \( 1 + (6.57 + 6.57i)T + 53iT^{2} \)
59 \( 1 + 9.90T + 59T^{2} \)
61 \( 1 + 4.21T + 61T^{2} \)
67 \( 1 + (-4.48 + 4.48i)T - 67iT^{2} \)
71 \( 1 + 6.16iT - 71T^{2} \)
73 \( 1 + (-3.06 + 3.06i)T - 73iT^{2} \)
79 \( 1 - 14.1T + 79T^{2} \)
83 \( 1 + (0.835 - 0.835i)T - 83iT^{2} \)
89 \( 1 - 4.55T + 89T^{2} \)
97 \( 1 + (3.92 - 3.92i)T - 97iT^{2} \)
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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

−9.562525280411059726311433185929, −9.102735385069216503057087678819, −8.069769948677875360815932172951, −6.84426991956621750560559957819, −6.55053717655873044315642829949, −6.13328149662252335779000720378, −4.52067760689828199371051171296, −3.61882939533437425374256656295, −1.62798498196143929949330545175, −0.56701470243500644530047384568, 1.30993564942130615703188209511, 3.02825250175732967946431966808, 3.80765601732886508717947042164, 5.00815357716528670868863371526, 5.97970812607144089116735511466, 6.64216977409079916768527301647, 7.983142955759974583107501268268, 8.965548044633024921674718245821, 9.420377373851081138285062568795, 10.32988613029912933489785609265

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