Properties

Label 2-95-95.74-c1-0-6
Degree $2$
Conductor $95$
Sign $0.735 + 0.677i$
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  + (0.240 + 0.0424i)2-s + (1.76 − 2.09i)3-s + (−1.82 − 0.663i)4-s + (0.0152 + 2.23i)5-s + (0.512 − 0.430i)6-s + (1.68 − 0.970i)7-s + (−0.834 − 0.481i)8-s + (−0.781 − 4.43i)9-s + (−0.0912 + 0.538i)10-s + (−2.11 + 3.66i)11-s + (−4.60 + 2.65i)12-s + (−0.816 − 0.972i)13-s + (0.445 − 0.162i)14-s + (4.71 + 3.90i)15-s + (2.79 + 2.34i)16-s + (2.42 + 0.427i)17-s + ⋯
L(s)  = 1  + (0.170 + 0.0300i)2-s + (1.01 − 1.21i)3-s + (−0.911 − 0.331i)4-s + (0.00680 + 0.999i)5-s + (0.209 − 0.175i)6-s + (0.635 − 0.366i)7-s + (−0.294 − 0.170i)8-s + (−0.260 − 1.47i)9-s + (−0.0288 + 0.170i)10-s + (−0.638 + 1.10i)11-s + (−1.32 + 0.766i)12-s + (−0.226 − 0.269i)13-s + (0.119 − 0.0433i)14-s + (1.21 + 1.00i)15-s + (0.698 + 0.585i)16-s + (0.587 + 0.103i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.735 + 0.677i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.735 + 0.677i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :1/2),\ 0.735 + 0.677i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12540 - 0.439363i\)
\(L(\frac12)\) \(\approx\) \(1.12540 - 0.439363i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.0152 - 2.23i)T \)
19 \( 1 + (1.64 - 4.03i)T \)
good2 \( 1 + (-0.240 - 0.0424i)T + (1.87 + 0.684i)T^{2} \)
3 \( 1 + (-1.76 + 2.09i)T + (-0.520 - 2.95i)T^{2} \)
7 \( 1 + (-1.68 + 0.970i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.11 - 3.66i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.816 + 0.972i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (-2.42 - 0.427i)T + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (1.75 - 4.82i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.50 + 8.50i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (2.55 + 4.41i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 11.0iT - 37T^{2} \)
41 \( 1 + (-1.91 - 1.60i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (1.52 + 4.19i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (-6.66 + 1.17i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (-0.235 + 0.648i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (0.901 - 5.11i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (2.63 + 0.958i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (5.13 - 0.905i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (0.744 - 0.270i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (0.910 - 1.08i)T + (-12.6 - 71.8i)T^{2} \)
79 \( 1 + (-4.64 - 3.89i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-10.2 + 5.92i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.71 + 1.43i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-8.47 - 1.49i)T + (91.1 + 33.1i)T^{2} \)
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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

−13.91386225775544826811483396596, −13.12178002288956446711758852996, −12.12838092504436124633135249094, −10.46860334623399200145882853299, −9.464857573292534940350432339251, −7.85504106738922442463997633006, −7.52445308055120856901568331017, −5.82988717924822473975363435509, −3.89673592305195156937670866018, −2.10092061793790731040691880711, 3.14907619985443487495839078083, 4.54604181712452225274566386439, 5.24808551932000291900300434018, 8.168932401764124655488652631315, 8.654438655368311962154059100851, 9.418340898312212895191890628347, 10.68990542175734688390575018461, 12.19866385576996180579269250988, 13.33301565598731924445286970970, 14.14837396698460932128539067519

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