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Properties

Label 2-845-845.389-c1-0-81
Degree $2$
Conductor $845$
Sign $-0.990 + 0.137i$
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  + (−0.545 − 0.483i)2-s + (1.32 + 0.502i)3-s + (−0.176 − 1.45i)4-s + (−2.16 − 0.563i)5-s + (−0.480 − 0.915i)6-s + (2.20 − 3.18i)7-s + (−1.43 + 2.08i)8-s + (−0.739 − 0.655i)9-s + (0.908 + 1.35i)10-s + (1.86 + 2.10i)11-s + (0.498 − 2.02i)12-s + (−3.59 − 0.229i)13-s + (−2.74 + 0.675i)14-s + (−2.58 − 1.83i)15-s + (−1.06 + 0.261i)16-s + (−3.80 − 2.62i)17-s + ⋯
L(s)  = 1  + (−0.385 − 0.341i)2-s + (0.765 + 0.290i)3-s + (−0.0884 − 0.728i)4-s + (−0.967 − 0.252i)5-s + (−0.196 − 0.373i)6-s + (0.831 − 1.20i)7-s + (−0.507 + 0.735i)8-s + (−0.246 − 0.218i)9-s + (0.287 + 0.428i)10-s + (0.562 + 0.634i)11-s + (0.143 − 0.583i)12-s + (−0.997 − 0.0635i)13-s + (−0.732 + 0.180i)14-s + (−0.667 − 0.474i)15-s + (−0.265 + 0.0653i)16-s + (−0.923 − 0.637i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.137i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.990 + 0.137i$
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: \((2,\ 845,\ (\ :1/2),\ -0.990 + 0.137i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0552216 - 0.797113i\)
\(L(\frac12)\) \(\approx\) \(0.0552216 - 0.797113i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.16 + 0.563i)T \)
13 \( 1 + (3.59 + 0.229i)T \)
good2 \( 1 + (0.545 + 0.483i)T + (0.241 + 1.98i)T^{2} \)
3 \( 1 + (-1.32 - 0.502i)T + (2.24 + 1.98i)T^{2} \)
7 \( 1 + (-2.20 + 3.18i)T + (-2.48 - 6.54i)T^{2} \)
11 \( 1 + (-1.86 - 2.10i)T + (-1.32 + 10.9i)T^{2} \)
17 \( 1 + (3.80 + 2.62i)T + (6.02 + 15.8i)T^{2} \)
19 \( 1 + 1.20iT - 19T^{2} \)
23 \( 1 - 1.21iT - 23T^{2} \)
29 \( 1 + (-3.04 - 2.69i)T + (3.49 + 28.7i)T^{2} \)
31 \( 1 + (4.17 + 7.95i)T + (-17.6 + 25.5i)T^{2} \)
37 \( 1 + (6.75 - 3.54i)T + (21.0 - 30.4i)T^{2} \)
41 \( 1 + (-6.69 - 2.53i)T + (30.6 + 27.1i)T^{2} \)
43 \( 1 + (4.45 - 8.48i)T + (-24.4 - 35.3i)T^{2} \)
47 \( 1 + (0.170 - 1.40i)T + (-45.6 - 11.2i)T^{2} \)
53 \( 1 + (11.8 + 8.18i)T + (18.7 + 49.5i)T^{2} \)
59 \( 1 + (1.07 - 4.37i)T + (-52.2 - 27.4i)T^{2} \)
61 \( 1 + (4.17 + 6.05i)T + (-21.6 + 57.0i)T^{2} \)
67 \( 1 + (-1.16 + 9.60i)T + (-65.0 - 16.0i)T^{2} \)
71 \( 1 + (-0.325 - 0.123i)T + (53.1 + 47.0i)T^{2} \)
73 \( 1 + (-9.87 + 8.74i)T + (8.79 - 72.4i)T^{2} \)
79 \( 1 + (-0.578 + 4.76i)T + (-76.7 - 18.9i)T^{2} \)
83 \( 1 + (3.12 + 8.24i)T + (-62.1 + 55.0i)T^{2} \)
89 \( 1 + 10.0iT - 89T^{2} \)
97 \( 1 + (-12.8 + 3.15i)T + (85.8 - 45.0i)T^{2} \)
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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

−9.577377714748509732793571136346, −9.189773508881450956995459835333, −8.150671490329581558689245520176, −7.50186104750580995938453792559, −6.53074710295646792206674452864, −4.81008232214427682424904251047, −4.50180746942309572363220766638, −3.23225760555587919047465547299, −1.84970310742372410342994182000, −0.38831902864886543315645486228, 2.16871031768529032600998125294, 3.09562174838535361278657545477, 4.11325671173488268077280509977, 5.29765889042113945681636385175, 6.64067511026797256433640060176, 7.45485198911720431651814615385, 8.245936905821295234124517826768, 8.647012996133917789497935311016, 9.160938803997364490760918294079, 10.72161764829137976595811744698

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