Properties

Label 2-845-169.114-c1-0-19
Degree $2$
Conductor $845$
Sign $-0.420 + 0.907i$
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  + (−2.02 − 1.65i)2-s + (0.230 + 0.0186i)3-s + (0.962 + 4.71i)4-s + (−0.992 + 0.120i)5-s + (−0.435 − 0.418i)6-s + (0.358 + 0.567i)7-s + (3.41 − 6.49i)8-s + (−2.90 − 0.472i)9-s + (2.20 + 1.39i)10-s + (0.526 + 3.23i)11-s + (0.134 + 1.10i)12-s + (−3.49 − 0.885i)13-s + (0.211 − 1.73i)14-s + (−0.231 + 0.00931i)15-s + (−8.77 + 3.73i)16-s + (4.10 − 2.59i)17-s + ⋯
L(s)  = 1  + (−1.42 − 1.16i)2-s + (0.133 + 0.0107i)3-s + (0.481 + 2.35i)4-s + (−0.443 + 0.0539i)5-s + (−0.177 − 0.170i)6-s + (0.135 + 0.214i)7-s + (1.20 − 2.29i)8-s + (−0.969 − 0.157i)9-s + (0.697 + 0.441i)10-s + (0.158 + 0.976i)11-s + (0.0387 + 0.318i)12-s + (−0.969 − 0.245i)13-s + (0.0564 − 0.464i)14-s + (−0.0596 + 0.00240i)15-s + (−2.19 + 0.934i)16-s + (0.994 − 0.629i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.420 + 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.420 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.420 + 0.907i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (621, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.420 + 0.907i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.270425 - 0.423638i\)
\(L(\frac12)\) \(\approx\) \(0.270425 - 0.423638i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.992 - 0.120i)T \)
13 \( 1 + (3.49 + 0.885i)T \)
good2 \( 1 + (2.02 + 1.65i)T + (0.400 + 1.95i)T^{2} \)
3 \( 1 + (-0.230 - 0.0186i)T + (2.96 + 0.481i)T^{2} \)
7 \( 1 + (-0.358 - 0.567i)T + (-3.00 + 6.32i)T^{2} \)
11 \( 1 + (-0.526 - 3.23i)T + (-10.4 + 3.48i)T^{2} \)
17 \( 1 + (-4.10 + 2.59i)T + (7.28 - 15.3i)T^{2} \)
19 \( 1 + (3.52 + 2.03i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.13 - 1.96i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.43 + 5.42i)T + (-5.80 - 28.4i)T^{2} \)
31 \( 1 + (2.62 - 10.6i)T + (-27.4 - 14.4i)T^{2} \)
37 \( 1 + (-7.20 + 2.08i)T + (31.2 - 19.7i)T^{2} \)
41 \( 1 + (-0.769 + 9.53i)T + (-40.4 - 6.57i)T^{2} \)
43 \( 1 + (-2.43 + 8.39i)T + (-36.3 - 22.9i)T^{2} \)
47 \( 1 + (-8.23 + 9.29i)T + (-5.66 - 46.6i)T^{2} \)
53 \( 1 + (-12.7 - 6.71i)T + (30.1 + 43.6i)T^{2} \)
59 \( 1 + (-1.49 + 3.50i)T + (-40.8 - 42.5i)T^{2} \)
61 \( 1 + (0.0381 - 0.945i)T + (-60.8 - 4.90i)T^{2} \)
67 \( 1 + (5.13 + 1.04i)T + (61.6 + 26.2i)T^{2} \)
71 \( 1 + (5.11 - 2.42i)T + (44.9 - 54.9i)T^{2} \)
73 \( 1 + (13.2 + 5.02i)T + (54.6 + 48.4i)T^{2} \)
79 \( 1 + (4.39 + 3.89i)T + (9.52 + 78.4i)T^{2} \)
83 \( 1 + (-1.00 + 0.695i)T + (29.4 - 77.6i)T^{2} \)
89 \( 1 + (-0.180 + 0.104i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.96 - 5.28i)T + (-26.9 - 93.1i)T^{2} \)
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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

−10.08349887055875297887889794811, −8.958135864309066809186604682552, −8.706188811050669473177097372495, −7.50864739322261530899567487624, −7.14625698241965608304241954080, −5.40360175244718780132199620453, −4.06664302160295967963384934285, −2.91243901268916299839795888432, −2.18685402039428210510768774865, −0.49457293563154414867525538150, 0.982319659215039746106065727352, 2.71921557287180016787945435612, 4.40597751886625317998670361083, 5.74499239500089677569716037769, 6.18412178655190508168071794025, 7.39936263749471841138181458368, 7.995504695539886639990944511582, 8.559859036254542755916914497578, 9.348994949272389445590819127646, 10.21757824371790881898544475596

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