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Properties

Label 2-950-95.8-c1-0-1
Degree $2$
Conductor $950$
Sign $-0.498 + 0.866i$
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 + (0.814 + 3.03i)3-s + (0.866 − 0.499i)4-s + (−1.57 − 2.72i)6-s + (−2.05 + 2.05i)7-s + (−0.707 + 0.707i)8-s + (−5.96 + 3.44i)9-s − 5.34·11-s + (2.22 + 2.22i)12-s + (3.75 + 1.00i)13-s + (1.45 − 2.51i)14-s + (0.500 − 0.866i)16-s + (−0.687 − 2.56i)17-s + (4.87 − 4.87i)18-s + (3.18 + 2.97i)19-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.469 + 1.75i)3-s + (0.433 − 0.249i)4-s + (−0.641 − 1.11i)6-s + (−0.777 + 0.777i)7-s + (−0.249 + 0.249i)8-s + (−1.98 + 1.14i)9-s − 1.61·11-s + (0.641 + 0.641i)12-s + (1.04 + 0.279i)13-s + (0.388 − 0.673i)14-s + (0.125 − 0.216i)16-s + (−0.166 − 0.622i)17-s + (1.14 − 1.14i)18-s + (0.730 + 0.682i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.498 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.498 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.498 + 0.866i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.498 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.280882 - 0.485648i\)
\(L(\frac12)\) \(\approx\) \(0.280882 - 0.485648i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.18 - 2.97i)T \)
good3 \( 1 + (-0.814 - 3.03i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (2.05 - 2.05i)T - 7iT^{2} \)
11 \( 1 + 5.34T + 11T^{2} \)
13 \( 1 + (-3.75 - 1.00i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (0.687 + 2.56i)T + (-14.7 + 8.5i)T^{2} \)
23 \( 1 + (0.752 - 2.80i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (3.53 + 6.11i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.91iT - 31T^{2} \)
37 \( 1 + (-3.67 - 3.67i)T + 37iT^{2} \)
41 \( 1 + (-4.61 - 2.66i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.51 - 1.74i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (2.32 + 0.622i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.42 - 0.917i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (4.58 - 7.93i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.199 - 0.345i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.00951 + 0.0355i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.91 - 2.26i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (15.5 - 4.16i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.15 - 1.99i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.45 - 3.45i)T + 83iT^{2} \)
89 \( 1 + (6.68 + 11.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-13.4 + 3.59i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.16744752018013702549625442828, −9.746741754446863836854678972098, −9.170157431596561781636971413318, −8.297935342571106232286896696789, −7.65416414152247772500468286585, −5.96238144153657084508571458197, −5.55934194972853580501249381129, −4.35796098988807319586778872987, −3.23204614270196475488259389381, −2.49294162801170582700972596531, 0.29992911792455873105414743411, 1.48690342197098386084952220112, 2.73019500040924103488428517314, 3.46135115879376690310442357549, 5.44575631912716061985203191932, 6.50442444486614695017927868440, 7.07976321599597575899036155309, 7.82683150692083363320930635694, 8.436620248426543118537895723936, 9.244494524527865806002196445676

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