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Properties

Label 2-950-95.12-c1-0-11
Degree $2$
Conductor $950$
Sign $0.847 + 0.530i$
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 + (0.186 − 0.694i)3-s + (0.866 + 0.499i)4-s + (−0.359 + 0.622i)6-s + (−1.01 − 1.01i)7-s + (−0.707 − 0.707i)8-s + (2.14 + 1.24i)9-s + 2.38·11-s + (0.508 − 0.508i)12-s + (2.90 − 0.778i)13-s + (0.717 + 1.24i)14-s + (0.500 + 0.866i)16-s + (−1.22 + 4.58i)17-s + (−1.75 − 1.75i)18-s + (−3.35 − 2.78i)19-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.107 − 0.401i)3-s + (0.433 + 0.249i)4-s + (−0.146 + 0.254i)6-s + (−0.383 − 0.383i)7-s + (−0.249 − 0.249i)8-s + (0.716 + 0.413i)9-s + 0.720·11-s + (0.146 − 0.146i)12-s + (0.805 − 0.215i)13-s + (0.191 + 0.331i)14-s + (0.125 + 0.216i)16-s + (−0.298 + 1.11i)17-s + (−0.413 − 0.413i)18-s + (−0.769 − 0.639i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.847 + 0.530i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.847 + 0.530i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24402 - 0.357325i\)
\(L(\frac12)\) \(\approx\) \(1.24402 - 0.357325i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.35 + 2.78i)T \)
good3 \( 1 + (-0.186 + 0.694i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (1.01 + 1.01i)T + 7iT^{2} \)
11 \( 1 - 2.38T + 11T^{2} \)
13 \( 1 + (-2.90 + 0.778i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (1.22 - 4.58i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (-1.35 - 5.07i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-4.88 + 8.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 10.0iT - 31T^{2} \)
37 \( 1 + (-2.87 + 2.87i)T - 37iT^{2} \)
41 \( 1 + (-6.97 + 4.02i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.76 + 0.471i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-0.349 + 0.0935i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-7.08 + 1.89i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.29 - 2.24i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.310 - 0.537i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.368 - 1.37i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (5.38 - 3.11i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.21 + 0.862i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.0163 + 0.0282i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-12.0 + 12.0i)T - 83iT^{2} \)
89 \( 1 + (-1.11 + 1.92i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.85 - 2.64i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.10608410102878007366426955531, −9.007428619202598781793822796829, −8.419539006174154671335259558784, −7.45643561142917317605239897859, −6.73022756945320761362094238535, −5.99618867244609630887958676626, −4.42624559462872926844901506248, −3.54836088302247688569906469747, −2.12907743351520472367436050425, −1.03105088102085826160419598494, 1.07046143185643693575435061869, 2.58028743416133277651361482563, 3.82579825745147070826356165027, 4.75462625177434280667246381424, 6.20203461155648847860166257858, 6.59860086686435010153571142085, 7.65537175719681782764511251112, 8.774022923552727410441260270019, 9.169852776054266185003977886511, 9.954851278965311526739470787895

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