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Properties

Label 2-845-845.389-c1-0-76
Degree $2$
Conductor $845$
Sign $-0.190 - 0.981i$
Analytic cond. $6.74735$
Root an. cond. $2.59756$
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  + (1.75 + 1.55i)2-s + (3.06 + 1.16i)3-s + (0.423 + 3.49i)4-s + (0.539 − 2.17i)5-s + (3.57 + 6.81i)6-s + (−0.871 + 1.26i)7-s + (−2.02 + 2.92i)8-s + (5.79 + 5.13i)9-s + (4.32 − 2.97i)10-s + (−2.53 − 2.86i)11-s + (−2.75 + 11.1i)12-s + (−3.60 − 0.154i)13-s + (−3.49 + 0.862i)14-s + (4.17 − 6.02i)15-s + (−1.29 + 0.318i)16-s + (−1.22 − 0.844i)17-s + ⋯
L(s)  = 1  + (1.24 + 1.10i)2-s + (1.76 + 0.670i)3-s + (0.211 + 1.74i)4-s + (0.241 − 0.970i)5-s + (1.46 + 2.78i)6-s + (−0.329 + 0.477i)7-s + (−0.714 + 1.03i)8-s + (1.93 + 1.71i)9-s + (1.36 − 0.940i)10-s + (−0.765 − 0.863i)11-s + (−0.795 + 3.22i)12-s + (−0.999 − 0.0427i)13-s + (−0.935 + 0.230i)14-s + (1.07 − 1.55i)15-s + (−0.322 + 0.0795i)16-s + (−0.296 − 0.204i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.190 - 0.981i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.190 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.16348 + 3.83502i\)
\(L(\frac12)\) \(\approx\) \(3.16348 + 3.83502i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.539 + 2.17i)T \)
13 \( 1 + (3.60 + 0.154i)T \)
good2 \( 1 + (-1.75 - 1.55i)T + (0.241 + 1.98i)T^{2} \)
3 \( 1 + (-3.06 - 1.16i)T + (2.24 + 1.98i)T^{2} \)
7 \( 1 + (0.871 - 1.26i)T + (-2.48 - 6.54i)T^{2} \)
11 \( 1 + (2.53 + 2.86i)T + (-1.32 + 10.9i)T^{2} \)
17 \( 1 + (1.22 + 0.844i)T + (6.02 + 15.8i)T^{2} \)
19 \( 1 + 7.90iT - 19T^{2} \)
23 \( 1 - 2.55iT - 23T^{2} \)
29 \( 1 + (7.56 + 6.70i)T + (3.49 + 28.7i)T^{2} \)
31 \( 1 + (-1.52 - 2.90i)T + (-17.6 + 25.5i)T^{2} \)
37 \( 1 + (4.06 - 2.13i)T + (21.0 - 30.4i)T^{2} \)
41 \( 1 + (-2.29 - 0.869i)T + (30.6 + 27.1i)T^{2} \)
43 \( 1 + (3.59 - 6.84i)T + (-24.4 - 35.3i)T^{2} \)
47 \( 1 + (0.858 - 7.07i)T + (-45.6 - 11.2i)T^{2} \)
53 \( 1 + (-2.16 - 1.49i)T + (18.7 + 49.5i)T^{2} \)
59 \( 1 + (0.200 - 0.813i)T + (-52.2 - 27.4i)T^{2} \)
61 \( 1 + (-5.71 - 8.28i)T + (-21.6 + 57.0i)T^{2} \)
67 \( 1 + (0.704 - 5.80i)T + (-65.0 - 16.0i)T^{2} \)
71 \( 1 + (-11.4 - 4.34i)T + (53.1 + 47.0i)T^{2} \)
73 \( 1 + (-3.38 + 2.99i)T + (8.79 - 72.4i)T^{2} \)
79 \( 1 + (0.0162 - 0.133i)T + (-76.7 - 18.9i)T^{2} \)
83 \( 1 + (2.27 + 5.99i)T + (-62.1 + 55.0i)T^{2} \)
89 \( 1 + 17.8iT - 89T^{2} \)
97 \( 1 + (-7.40 + 1.82i)T + (85.8 - 45.0i)T^{2} \)
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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

−9.984505972754832789800424623175, −9.283207077363885034284446239052, −8.615187644102939197511090877206, −7.85635925492844562343036751365, −7.15838546293881435507908026536, −5.77389424236437372307324022282, −4.91539103959612092122662168435, −4.36103745473688182626073134300, −3.17038942913044743699063660573, −2.46560938154114809347654956867, 2.01536131556866206733658522459, 2.22886416010441589533825699525, 3.49067166247750328734709096355, 3.81065379691344198114124707085, 5.24944563804007280125886503692, 6.63571971864412213766961216201, 7.34128204658498654656343301735, 8.100763663467191320440870714569, 9.497624817190313794029725297526, 10.11303425507591991380230760314

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