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Properties

Label 2-845-65.4-c1-0-3
Degree $2$
Conductor $845$
Sign $-0.986 - 0.165i$
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  + (−0.769 + 1.33i)2-s + (−2.74 − 1.58i)3-s + (−0.184 − 0.319i)4-s + (2.17 + 0.539i)5-s + (4.22 − 2.43i)6-s + (−0.854 − 1.48i)7-s − 2.51·8-s + (3.52 + 6.10i)9-s + (−2.38 + 2.47i)10-s + (−2.19 − 1.26i)11-s + 1.17i·12-s + 2.63·14-s + (−5.10 − 4.91i)15-s + (2.30 − 3.98i)16-s + (0.798 − 0.460i)17-s − 10.8·18-s + ⋯
L(s)  = 1  + (−0.544 + 0.942i)2-s + (−1.58 − 0.915i)3-s + (−0.0922 − 0.159i)4-s + (0.970 + 0.241i)5-s + (1.72 − 0.995i)6-s + (−0.323 − 0.559i)7-s − 0.887·8-s + (1.17 + 2.03i)9-s + (−0.755 + 0.783i)10-s + (−0.663 − 0.382i)11-s + 0.337i·12-s + 0.703·14-s + (−1.31 − 1.27i)15-s + (0.575 − 0.996i)16-s + (0.193 − 0.111i)17-s − 2.55·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.165i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.986 - 0.165i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (654, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.986 - 0.165i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0216755 + 0.260070i\)
\(L(\frac12)\) \(\approx\) \(0.0216755 + 0.260070i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.17 - 0.539i)T \)
13 \( 1 \)
good2 \( 1 + (0.769 - 1.33i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (2.74 + 1.58i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (0.854 + 1.48i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.19 + 1.26i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.798 + 0.460i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.466 + 0.269i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.45 - 1.41i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.56 - 4.44i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.879iT - 31T^{2} \)
37 \( 1 + (3.02 - 5.23i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.09 + 0.630i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.57 - 3.21i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.70T + 47T^{2} \)
53 \( 1 - 8.49iT - 53T^{2} \)
59 \( 1 + (4.08 - 2.36i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.02 + 6.97i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.93 - 6.81i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (12.5 - 7.24i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.95T + 73T^{2} \)
79 \( 1 + 0.496T + 79T^{2} \)
83 \( 1 + 8.63T + 83T^{2} \)
89 \( 1 + (11.1 + 6.41i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.95 + 5.12i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.54915285380948963379121363553, −9.875987942515710560558526727423, −8.718382503256202876425113315263, −7.61003472051837990423979940771, −7.02505458236853321937948273886, −6.41893442261412473109939093716, −5.68525917528151047053484839875, −5.03434472373759935756475738191, −2.98926848746696963183225079686, −1.34218711657373796781727985802, 0.19450798317359261294366957117, 1.72911563668004761609646242691, 3.06923125174859647602160561262, 4.53166157481278840573582053710, 5.54523086590474635520954625021, 5.89497905296924954572213692381, 6.87581668886878569506627171859, 8.637351055285475277656611070353, 9.592979509132484698951373501465, 9.842229577100907920271102179731

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