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Properties

Label 2-595-119.16-c1-0-30
Degree $2$
Conductor $595$
Sign $0.608 - 0.793i$
Analytic cond. $4.75109$
Root an. cond. $2.17970$
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  + (−1.27 + 2.21i)2-s + (2.06 − 1.19i)3-s + (−2.25 − 3.91i)4-s + (0.866 + 0.5i)5-s + 6.09i·6-s + (2.39 − 1.12i)7-s + 6.41·8-s + (1.35 − 2.34i)9-s + (−2.21 + 1.27i)10-s + (−3.62 + 2.09i)11-s + (−9.33 − 5.39i)12-s + 4.01·13-s + (−0.580 + 6.72i)14-s + 2.38·15-s + (−3.67 + 6.37i)16-s + (0.441 − 4.09i)17-s + ⋯
L(s)  = 1  + (−0.902 + 1.56i)2-s + (1.19 − 0.689i)3-s + (−1.12 − 1.95i)4-s + (0.387 + 0.223i)5-s + 2.48i·6-s + (0.905 − 0.423i)7-s + 2.26·8-s + (0.450 − 0.780i)9-s + (−0.699 + 0.403i)10-s + (−1.09 + 0.630i)11-s + (−2.69 − 1.55i)12-s + 1.11·13-s + (−0.155 + 1.79i)14-s + 0.616·15-s + (−0.919 + 1.59i)16-s + (0.107 − 0.994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(595\)    =    \(5 \cdot 7 \cdot 17\)
Sign: $0.608 - 0.793i$
Analytic conductor: \(4.75109\)
Root analytic conductor: \(2.17970\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{595} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 595,\ (\ :1/2),\ 0.608 - 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34598 + 0.664523i\)
\(L(\frac12)\) \(\approx\) \(1.34598 + 0.664523i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-2.39 + 1.12i)T \)
17 \( 1 + (-0.441 + 4.09i)T \)
good2 \( 1 + (1.27 - 2.21i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-2.06 + 1.19i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (3.62 - 2.09i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.01T + 13T^{2} \)
19 \( 1 + (-1.00 + 1.74i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.480 - 0.277i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.78iT - 29T^{2} \)
31 \( 1 + (-5.36 + 3.09i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.18 + 0.685i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.93iT - 41T^{2} \)
43 \( 1 - 11.3T + 43T^{2} \)
47 \( 1 + (4.97 - 8.61i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.87 + 10.1i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.72 - 8.18i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.03 + 0.595i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.38 - 4.13i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.13iT - 71T^{2} \)
73 \( 1 + (14.1 - 8.18i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.84 + 4.52i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 + (-3.57 + 6.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.47iT - 97T^{2} \)
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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

−10.36214036688596324910958367852, −9.511420367063456969745017721852, −8.714229410422598823125365806747, −7.927364108560550343755735699768, −7.56158926802561035336400702009, −6.74653890083687761016889358308, −5.62145220084230122238588388531, −4.58967304107065354224958172408, −2.66199755973350796804744195928, −1.24481979296059573585081232615, 1.45091792275739861961612289491, 2.54757675186091818277930797195, 3.39934071738102319141571567232, 4.36967252074247898104482875211, 5.71497315309784172936891048925, 7.84721049792467445919130256641, 8.524843519027117452110789937589, 8.718717653843917553977183525975, 9.745697526861986303447705217565, 10.54994843681022378431855883131

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