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Properties

Label 2-845-65.8-c1-0-66
Degree $2$
Conductor $845$
Sign $-0.823 + 0.566i$
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  + 1.83·2-s + (−1.40 − 1.40i)3-s + 1.35·4-s + (1.45 − 1.69i)5-s + (−2.56 − 2.56i)6-s − 3.53i·7-s − 1.18·8-s + 0.927i·9-s + (2.66 − 3.11i)10-s + (−2.73 + 2.73i)11-s + (−1.89 − 1.89i)12-s − 6.48i·14-s + (−4.41 + 0.340i)15-s − 4.87·16-s + (1.43 + 1.43i)17-s + 1.69i·18-s + ⋯
L(s)  = 1  + 1.29·2-s + (−0.809 − 0.809i)3-s + 0.677·4-s + (0.650 − 0.759i)5-s + (−1.04 − 1.04i)6-s − 1.33i·7-s − 0.417·8-s + 0.309i·9-s + (0.842 − 0.983i)10-s + (−0.825 + 0.825i)11-s + (−0.548 − 0.548i)12-s − 1.73i·14-s + (−1.14 + 0.0880i)15-s − 1.21·16-s + (0.347 + 0.347i)17-s + 0.400i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.823 + 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.823 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.823 + 0.566i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.823 + 0.566i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.564011 - 1.81417i\)
\(L(\frac12)\) \(\approx\) \(0.564011 - 1.81417i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.45 + 1.69i)T \)
13 \( 1 \)
good2 \( 1 - 1.83T + 2T^{2} \)
3 \( 1 + (1.40 + 1.40i)T + 3iT^{2} \)
7 \( 1 + 3.53iT - 7T^{2} \)
11 \( 1 + (2.73 - 2.73i)T - 11iT^{2} \)
17 \( 1 + (-1.43 - 1.43i)T + 17iT^{2} \)
19 \( 1 + (-0.379 + 0.379i)T - 19iT^{2} \)
23 \( 1 + (-0.215 + 0.215i)T - 23iT^{2} \)
29 \( 1 + 1.97iT - 29T^{2} \)
31 \( 1 + (-4.13 - 4.13i)T + 31iT^{2} \)
37 \( 1 + 5.41iT - 37T^{2} \)
41 \( 1 + (0.475 + 0.475i)T + 41iT^{2} \)
43 \( 1 + (-6.23 + 6.23i)T - 43iT^{2} \)
47 \( 1 + 9.75iT - 47T^{2} \)
53 \( 1 + (-3.16 - 3.16i)T + 53iT^{2} \)
59 \( 1 + (8.59 + 8.59i)T + 59iT^{2} \)
61 \( 1 + 2.88T + 61T^{2} \)
67 \( 1 - 2.28T + 67T^{2} \)
71 \( 1 + (3.26 + 3.26i)T + 71iT^{2} \)
73 \( 1 - 14.7T + 73T^{2} \)
79 \( 1 + 1.59iT - 79T^{2} \)
83 \( 1 + 7.57iT - 83T^{2} \)
89 \( 1 + (-3.32 - 3.32i)T + 89iT^{2} \)
97 \( 1 - 17.8T + 97T^{2} \)
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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

−10.10841066819790611761000239247, −9.073411345227824291064189882966, −7.76983522793911822656714315078, −6.95988227360251005333945378302, −6.17524345240041718586346519859, −5.33443037819133225819576748687, −4.66748915890117655397231218912, −3.67790424158164299568932773513, −2.07974022916678249273426651688, −0.64916717224313389081073270238, 2.51905980455438936025918314160, 3.14815668023792676703251790990, 4.51605843157704044167844062119, 5.38170580635047211558105690697, 5.79263736551601276492120361860, 6.41176109993559223511813439597, 7.912406311879357519316077546580, 9.112905739641439516461960952404, 9.840709090226788300227259647899, 10.80281553120380373866962913905

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