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Properties

Label 2-845-65.29-c1-0-24
Degree $2$
Conductor $845$
Sign $0.998 - 0.0571i$
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  + (−0.342 + 0.197i)2-s + (−2.36 + 1.36i)3-s + (−0.921 + 1.59i)4-s + (−1.15 + 1.91i)5-s + (0.540 − 0.936i)6-s + (0.545 + 0.314i)7-s − 1.51i·8-s + (2.24 − 3.88i)9-s + (0.0151 − 0.883i)10-s + (−2.74 − 4.76i)11-s − 5.04i·12-s − 0.248·14-s + (0.105 − 6.11i)15-s + (−1.54 − 2.67i)16-s + (3.68 + 2.12i)17-s + 1.77i·18-s + ⋯
L(s)  = 1  + (−0.241 + 0.139i)2-s + (−1.36 + 0.789i)3-s + (−0.460 + 0.798i)4-s + (−0.514 + 0.857i)5-s + (0.220 − 0.382i)6-s + (0.206 + 0.118i)7-s − 0.536i·8-s + (0.748 − 1.29i)9-s + (0.00480 − 0.279i)10-s + (−0.828 − 1.43i)11-s − 1.45i·12-s − 0.0664·14-s + (0.0271 − 1.57i)15-s + (−0.385 − 0.668i)16-s + (0.893 + 0.515i)17-s + 0.417i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0571i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.998 - 0.0571i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (484, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.998 - 0.0571i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.344118 + 0.00984008i\)
\(L(\frac12)\) \(\approx\) \(0.344118 + 0.00984008i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.15 - 1.91i)T \)
13 \( 1 \)
good2 \( 1 + (0.342 - 0.197i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (2.36 - 1.36i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-0.545 - 0.314i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.74 + 4.76i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.68 - 2.12i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.80 + 3.13i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.11 - 3.52i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.04 - 1.81i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.97T + 31T^{2} \)
37 \( 1 + (0.708 - 0.408i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.44 - 4.23i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.58 + 4.95i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.29iT - 47T^{2} \)
53 \( 1 - 5.84iT - 53T^{2} \)
59 \( 1 + (-2.59 + 4.49i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.96 - 5.13i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.8 + 6.85i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.105 - 0.182i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 12.4iT - 73T^{2} \)
79 \( 1 - 7.19T + 79T^{2} \)
83 \( 1 + 6.61iT - 83T^{2} \)
89 \( 1 + (-1.83 - 3.17i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.93 + 2.27i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.34472033880407874031449826156, −9.582992972463469624188853664681, −8.308947892454163485646150124396, −7.80374426588350437791341718535, −6.64037491850146155894811876029, −5.77195700613306053745510530805, −4.92174704805239577495894407717, −3.80293671790452844614484535961, −3.09980990568279263335980316848, −0.30952098084120458017451833240, 0.892199189783671986266758916275, 1.94653120501023771059287626935, 4.28427402204874780950950134769, 5.09233229172015671077486622101, 5.57703448049459129148701900899, 6.66564665145373503864640406209, 7.70036297472135023658305932505, 8.261332220161727164447742710741, 9.738595897563479837266122104429, 10.08220032227644128745192140638

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