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Properties

Label 2-845-13.9-c1-0-48
Degree $2$
Conductor $845$
Sign $-0.945 - 0.326i$
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  + (1.24 − 2.16i)2-s + (1.41 − 2.44i)3-s + (−2.11 − 3.66i)4-s + 5-s + (−3.52 − 6.10i)6-s + (0.952 + 1.64i)7-s − 5.55·8-s + (−2.49 − 4.32i)9-s + (1.24 − 2.16i)10-s + (0.534 − 0.926i)11-s − 11.9·12-s + 4.75·14-s + (1.41 − 2.44i)15-s + (−2.70 + 4.69i)16-s + (−0.318 − 0.551i)17-s − 12.4·18-s + ⋯
L(s)  = 1  + (0.882 − 1.52i)2-s + (0.816 − 1.41i)3-s + (−1.05 − 1.83i)4-s + 0.447·5-s + (−1.43 − 2.49i)6-s + (0.360 + 0.623i)7-s − 1.96·8-s + (−0.831 − 1.44i)9-s + (0.394 − 0.683i)10-s + (0.161 − 0.279i)11-s − 3.44·12-s + 1.27·14-s + (0.364 − 0.632i)15-s + (−0.677 + 1.17i)16-s + (−0.0772 − 0.133i)17-s − 2.93·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.945 - 0.326i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.945 - 0.326i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.546675 + 3.25677i\)
\(L(\frac12)\) \(\approx\) \(0.546675 + 3.25677i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
13 \( 1 \)
good2 \( 1 + (-1.24 + 2.16i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.41 + 2.44i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-0.952 - 1.64i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.534 + 0.926i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.318 + 0.551i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.86 - 4.96i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.90 - 3.30i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.72 - 8.18i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.46T + 31T^{2} \)
37 \( 1 + (-0.378 + 0.655i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.133 + 0.232i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.318 + 0.551i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.44T + 47T^{2} \)
53 \( 1 + 6.99T + 53T^{2} \)
59 \( 1 + (-0.370 - 0.641i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.09 + 3.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.04 + 7.01i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.88 - 8.45i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 3.71T + 73T^{2} \)
79 \( 1 + 9.31T + 79T^{2} \)
83 \( 1 + 5.11T + 83T^{2} \)
89 \( 1 + (6.28 - 10.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.11 + 3.65i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.769678264821375511492831886184, −9.057223200353260327516105749015, −8.167406363271889187735620272328, −7.15031978830180353911576155081, −5.92805771432015822491495846720, −5.27537847837415655477305569836, −3.75788023557850010252427638885, −2.90554181022745163427847394903, −1.96389422138464514973938544498, −1.30388454774815225141967179358, 2.69525534628405342916649396025, 3.92047663743441391393373328231, 4.43707968915906208272078157396, 5.20396491525225508579743134692, 6.18696058640016645798245621411, 7.25868171911933872364141324685, 7.999179395049332112413441267741, 8.865209159855598678676656607638, 9.549432520533781717214990423575, 10.40099233359662855845456832463

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