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Properties

Label 2-95-19.16-c1-0-5
Degree $2$
Conductor $95$
Sign $-0.496 + 0.867i$
Analytic cond. $0.758578$
Root an. cond. $0.870964$
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  + (−0.484 − 0.406i)2-s + (−1.80 + 0.656i)3-s + (−0.277 − 1.57i)4-s + (0.173 − 0.984i)5-s + (1.14 + 0.415i)6-s + (−2.04 − 3.54i)7-s + (−1.13 + 1.97i)8-s + (0.524 − 0.440i)9-s + (−0.484 + 0.406i)10-s + (2.17 − 3.76i)11-s + (1.53 + 2.65i)12-s + (−1.45 − 0.530i)13-s + (−0.449 + 2.54i)14-s + (0.333 + 1.89i)15-s + (−1.65 + 0.601i)16-s + (4.87 + 4.08i)17-s + ⋯
L(s)  = 1  + (−0.342 − 0.287i)2-s + (−1.04 + 0.379i)3-s + (−0.138 − 0.787i)4-s + (0.0776 − 0.440i)5-s + (0.465 + 0.169i)6-s + (−0.772 − 1.33i)7-s + (−0.402 + 0.697i)8-s + (0.174 − 0.146i)9-s + (−0.153 + 0.128i)10-s + (0.655 − 1.13i)11-s + (0.443 + 0.767i)12-s + (−0.404 − 0.147i)13-s + (−0.120 + 0.681i)14-s + (0.0860 + 0.488i)15-s + (−0.412 + 0.150i)16-s + (1.18 + 0.991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.496 + 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.496 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $-0.496 + 0.867i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :1/2),\ -0.496 + 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.241403 - 0.416307i\)
\(L(\frac12)\) \(\approx\) \(0.241403 - 0.416307i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (-0.708 - 4.30i)T \)
good2 \( 1 + (0.484 + 0.406i)T + (0.347 + 1.96i)T^{2} \)
3 \( 1 + (1.80 - 0.656i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (2.04 + 3.54i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.17 + 3.76i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.45 + 0.530i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-4.87 - 4.08i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (0.583 + 3.31i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-3.99 + 3.35i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (3.28 + 5.68i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.180T + 37T^{2} \)
41 \( 1 + (0.0242 - 0.00881i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.793 + 4.50i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (1.09 - 0.919i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (0.278 + 1.57i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-7.31 - 6.13i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.05 + 5.99i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-7.87 + 6.60i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.88 - 10.7i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-12.7 + 4.65i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (16.2 - 5.90i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-2.57 - 4.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.477 - 0.173i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-2.51 - 2.10i)T + (16.8 + 95.5i)T^{2} \)
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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

−13.72986325017835878840730690450, −12.36578615795970499324115514702, −11.23041714154790723298094942724, −10.33868555714508002584855250323, −9.795549132417880960142245049089, −8.228371739686140341258604136133, −6.34398234327662814713344059147, −5.55135317589676692776915918797, −3.96869492712991728507848203525, −0.74573245551495280457099845822, 3.01642687849450408067427699281, 5.20751095781043555153913113053, 6.57261574899617416363508975052, 7.26886099829141508134747033187, 9.002189238852360506925575140452, 9.768145554685064255160897422110, 11.61979456790037001369786930601, 12.14259154853835384423536061838, 12.82908819145219546753340419823, 14.45982886296772958340197897680

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