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Properties

Label 2-845-65.4-c1-0-10
Degree $2$
Conductor $845$
Sign $-0.543 - 0.839i$
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  + (−0.165 + 0.286i)2-s + (−2.33 − 1.34i)3-s + (0.945 + 1.63i)4-s + (0.702 + 2.12i)5-s + (0.771 − 0.445i)6-s + (1.67 + 2.90i)7-s − 1.28·8-s + (2.12 + 3.67i)9-s + (−0.724 − 0.149i)10-s + (2.81 + 1.62i)11-s − 5.08i·12-s − 1.10·14-s + (1.21 − 5.89i)15-s + (−1.67 + 2.90i)16-s + (1.68 − 0.974i)17-s − 1.40·18-s + ⋯
L(s)  = 1  + (−0.116 + 0.202i)2-s + (−1.34 − 0.777i)3-s + (0.472 + 0.818i)4-s + (0.314 + 0.949i)5-s + (0.314 − 0.181i)6-s + (0.633 + 1.09i)7-s − 0.455·8-s + (0.707 + 1.22i)9-s + (−0.229 − 0.0474i)10-s + (0.847 + 0.489i)11-s − 1.46i·12-s − 0.296·14-s + (0.314 − 1.52i)15-s + (−0.419 + 0.726i)16-s + (0.409 − 0.236i)17-s − 0.331·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.543 - 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.543 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.543 - 0.839i$
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: \((2,\ 845,\ (\ :1/2),\ -0.543 - 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.499359 + 0.918534i\)
\(L(\frac12)\) \(\approx\) \(0.499359 + 0.918534i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.702 - 2.12i)T \)
13 \( 1 \)
good2 \( 1 + (0.165 - 0.286i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (2.33 + 1.34i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1.67 - 2.90i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.81 - 1.62i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.68 + 0.974i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.07 - 0.622i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.33 + 1.34i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.78iT - 31T^{2} \)
37 \( 1 + (-0.974 + 1.68i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.40 - 1.39i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (7.56 - 4.36i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.86T + 47T^{2} \)
53 \( 1 - 12.8iT - 53T^{2} \)
59 \( 1 + (-2.19 + 1.26i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.74 - 6.48i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.00 + 3.47i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.54 - 2.62i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 5.46T + 73T^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 + 8.61T + 83T^{2} \)
89 \( 1 + (-8.93 - 5.15i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.63 - 4.56i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.81553515169519230427661932340, −9.723173562789707396146031130723, −8.596935430408487101336424615915, −7.60919740368321857582766058652, −6.98297231658686905463709159873, −6.14386185753283781822241765021, −5.71542635966323116792528957935, −4.28202176107168100189649485578, −2.72630502742075595477157178954, −1.75542912930147761176695543505, 0.65767443876114399391810723738, 1.56980882749215849084453060606, 3.81768001230092165528496749956, 4.72452328362808994461846977154, 5.41507027051197905218812787632, 6.17505643427860922530054096985, 7.03484101512069808334164976985, 8.414262837092078242324373693914, 9.387949035879890744412938531297, 10.26841107772784420794139749220

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