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Properties

Label 2-845-13.12-c1-0-32
Degree $2$
Conductor $845$
Sign $0.969 - 0.246i$
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  + 2.20i·2-s − 0.0130·3-s − 2.86·4-s + i·5-s − 0.0287i·6-s − 4.60i·7-s − 1.90i·8-s − 2.99·9-s − 2.20·10-s − 2.93i·11-s + 0.0374·12-s + 10.1·14-s − 0.0130i·15-s − 1.52·16-s + 3.35·17-s − 6.61i·18-s + ⋯
L(s)  = 1  + 1.55i·2-s − 0.00753·3-s − 1.43·4-s + 0.447i·5-s − 0.0117i·6-s − 1.74i·7-s − 0.674i·8-s − 0.999·9-s − 0.697·10-s − 0.884i·11-s + 0.0107·12-s + 2.71·14-s − 0.00337i·15-s − 0.380·16-s + 0.812·17-s − 1.55i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.246i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 - 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.969 - 0.246i$
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: \((2,\ 845,\ (\ :1/2),\ 0.969 - 0.246i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06292 + 0.133261i\)
\(L(\frac12)\) \(\approx\) \(1.06292 + 0.133261i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.20iT - 2T^{2} \)
3 \( 1 + 0.0130T + 3T^{2} \)
7 \( 1 + 4.60iT - 7T^{2} \)
11 \( 1 + 2.93iT - 11T^{2} \)
17 \( 1 - 3.35T + 17T^{2} \)
19 \( 1 + 2.46iT - 19T^{2} \)
23 \( 1 - 1.58T + 23T^{2} \)
29 \( 1 - 8.26T + 29T^{2} \)
31 \( 1 + 9.77iT - 31T^{2} \)
37 \( 1 + 4.12iT - 37T^{2} \)
41 \( 1 - 7.26iT - 41T^{2} \)
43 \( 1 - 0.705T + 43T^{2} \)
47 \( 1 + 8.57iT - 47T^{2} \)
53 \( 1 + 12.4T + 53T^{2} \)
59 \( 1 - 5.33iT - 59T^{2} \)
61 \( 1 - 1.92T + 61T^{2} \)
67 \( 1 + 7.29iT - 67T^{2} \)
71 \( 1 + 6.68iT - 71T^{2} \)
73 \( 1 - 12.4iT - 73T^{2} \)
79 \( 1 - 0.984T + 79T^{2} \)
83 \( 1 + 7.84iT - 83T^{2} \)
89 \( 1 - 0.412iT - 89T^{2} \)
97 \( 1 + 3.38iT - 97T^{2} \)
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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

−10.13103960684976722285079193637, −9.119070766516823755177543144911, −8.082810191230466179023164053866, −7.69863557166927843140736764143, −6.71825284205807791422257030623, −6.15124098874108909571114081545, −5.15492043735575689995453850075, −4.10717066967000434871968644005, −2.99546505666188380499717542609, −0.54487493305297188036147423628, 1.48742495522701676805496851997, 2.58109827318647139008446015719, 3.26056344289801659279019291222, 4.73789650581403521934334960505, 5.41004063077052291133435575033, 6.49961977927353976406093032762, 8.114068295809543194559286927812, 8.792047924031978743302881353673, 9.422219425930450089257641884282, 10.20276456004275451711823841675

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