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Properties

Label 2-95-95.12-c1-0-7
Degree $2$
Conductor $95$
Sign $-0.0923 + 0.995i$
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  + (−0.349 − 0.0935i)2-s + (0.309 − 1.15i)3-s + (−1.61 − 0.934i)4-s + (−1.05 − 1.96i)5-s + (−0.216 + 0.374i)6-s + (−0.933 − 0.933i)7-s + (0.988 + 0.988i)8-s + (1.35 + 0.783i)9-s + (0.185 + 0.786i)10-s + 2.12·11-s + (−1.58 + 1.58i)12-s + (0.997 − 0.267i)13-s + (0.238 + 0.412i)14-s + (−2.60 + 0.615i)15-s + (1.61 + 2.80i)16-s + (1.63 − 6.12i)17-s + ⋯
L(s)  = 1  + (−0.246 − 0.0661i)2-s + (0.178 − 0.667i)3-s + (−0.809 − 0.467i)4-s + (−0.473 − 0.880i)5-s + (−0.0882 + 0.152i)6-s + (−0.352 − 0.352i)7-s + (0.349 + 0.349i)8-s + (0.452 + 0.261i)9-s + (0.0587 + 0.248i)10-s + 0.639·11-s + (−0.456 + 0.456i)12-s + (0.276 − 0.0741i)13-s + (0.0637 + 0.110i)14-s + (−0.672 + 0.158i)15-s + (0.404 + 0.700i)16-s + (0.397 − 1.48i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0923 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0923 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $-0.0923 + 0.995i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :1/2),\ -0.0923 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.510496 - 0.560062i\)
\(L(\frac12)\) \(\approx\) \(0.510496 - 0.560062i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.05 + 1.96i)T \)
19 \( 1 + (-0.0230 - 4.35i)T \)
good2 \( 1 + (0.349 + 0.0935i)T + (1.73 + i)T^{2} \)
3 \( 1 + (-0.309 + 1.15i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (0.933 + 0.933i)T + 7iT^{2} \)
11 \( 1 - 2.12T + 11T^{2} \)
13 \( 1 + (-0.997 + 0.267i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-1.63 + 6.12i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (-0.0329 - 0.123i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (1.30 - 2.26i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.35iT - 31T^{2} \)
37 \( 1 + (2.80 - 2.80i)T - 37iT^{2} \)
41 \( 1 + (-10.1 + 5.83i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.90 + 0.777i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (6.01 - 1.61i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-8.80 + 2.35i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.0799 + 0.138i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.21 - 7.29i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.65 - 13.6i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (4.19 - 2.42i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.830 + 0.222i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.80 - 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.0 - 10.0i)T - 83iT^{2} \)
89 \( 1 + (-4.77 + 8.27i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-17.4 - 4.68i)T + (84.0 + 48.5i)T^{2} \)
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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

−13.61122051873961979901448097898, −12.80593839973783718188141714500, −11.80714767036715405490028224935, −10.25634526129130480448490191706, −9.254821805453500512543538489198, −8.228475791364174806856619716470, −7.10536694717180748667044647009, −5.34848002421297891478744542397, −4.02868584348127549440355157176, −1.16782118955669905387441799440, 3.41619809885386439246942231904, 4.32551152739645304328092867892, 6.35399986693829064156751618277, 7.71101238096082473466903565618, 8.946920372387925065363221925659, 9.794934963768317532604738468972, 10.88346281629448656067242204163, 12.21162530840632334054422589289, 13.21439232029184824398565815839, 14.50547454834807714687662053717

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