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Properties

Label 2-845-13.12-c1-0-23
Degree $2$
Conductor $845$
Sign $-0.722 - 0.691i$
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  + 2.28i·2-s + 3.21·3-s − 3.20·4-s − i·5-s + 7.33i·6-s + 2.30i·7-s − 2.74i·8-s + 7.33·9-s + 2.28·10-s + 1.25i·11-s − 10.2·12-s − 5.25·14-s − 3.21i·15-s − 0.151·16-s − 2.43·17-s + 16.7i·18-s + ⋯
L(s)  = 1  + 1.61i·2-s + 1.85·3-s − 1.60·4-s − 0.447i·5-s + 2.99i·6-s + 0.870i·7-s − 0.969i·8-s + 2.44·9-s + 0.721·10-s + 0.379i·11-s − 2.97·12-s − 1.40·14-s − 0.829i·15-s − 0.0378·16-s − 0.589·17-s + 3.94i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.722 - 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.722 - 0.691i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (506, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.722 - 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02273 + 2.54639i\)
\(L(\frac12)\) \(\approx\) \(1.02273 + 2.54639i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + iT \)
13 \( 1 \)
good2 \( 1 - 2.28iT - 2T^{2} \)
3 \( 1 - 3.21T + 3T^{2} \)
7 \( 1 - 2.30iT - 7T^{2} \)
11 \( 1 - 1.25iT - 11T^{2} \)
17 \( 1 + 2.43T + 17T^{2} \)
19 \( 1 + 0.586iT - 19T^{2} \)
23 \( 1 - 8.37T + 23T^{2} \)
29 \( 1 + 3.09T + 29T^{2} \)
31 \( 1 + 0.394iT - 31T^{2} \)
37 \( 1 + 2.01iT - 37T^{2} \)
41 \( 1 - 5.58iT - 41T^{2} \)
43 \( 1 + 5.87T + 43T^{2} \)
47 \( 1 + 1.92iT - 47T^{2} \)
53 \( 1 + 13.6T + 53T^{2} \)
59 \( 1 + 14.6iT - 59T^{2} \)
61 \( 1 - 5.85T + 61T^{2} \)
67 \( 1 + 1.99iT - 67T^{2} \)
71 \( 1 - 9.59iT - 71T^{2} \)
73 \( 1 + 4.38iT - 73T^{2} \)
79 \( 1 + 0.217T + 79T^{2} \)
83 \( 1 + 11.0iT - 83T^{2} \)
89 \( 1 - 6.37iT - 89T^{2} \)
97 \( 1 + 16.3iT - 97T^{2} \)
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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

−9.770683069801725791424682066459, −9.156672805833702187574526511053, −8.687293975967155440505996607243, −8.039549539407296892472433847298, −7.25732918992148459212275994260, −6.50280592462567828879635949247, −5.18545653148604178195217372612, −4.46117475981463985460223794656, −3.17163359158948156581992355855, −1.98255837207307797673504128723, 1.25341889123585495711550298723, 2.37913355051274459718087735365, 3.20437983674711761707234032089, 3.80006678037405610857626578215, 4.71761414182019752295250345028, 6.79994547102726963359681811105, 7.52968678439724767510460522741, 8.598051846491594682012280185139, 9.171752772139051604000572909355, 9.932920532253430867550898232192

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