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Properties

Label 2-845-65.58-c1-0-51
Degree $2$
Conductor $845$
Sign $0.438 + 0.898i$
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  + (1.36 + 0.789i)2-s + (−0.991 + 0.265i)3-s + (0.247 + 0.428i)4-s + (−0.146 + 2.23i)5-s + (−1.56 − 0.419i)6-s + (−2.12 − 3.68i)7-s − 2.37i·8-s + (−1.68 + 0.973i)9-s + (−1.96 + 2.93i)10-s + (1.56 − 0.419i)11-s + (−0.358 − 0.358i)12-s − 6.71i·14-s + (−0.447 − 2.25i)15-s + (2.37 − 4.10i)16-s + (1.60 − 5.98i)17-s − 3.07·18-s + ⋯
L(s)  = 1  + (0.967 + 0.558i)2-s + (−0.572 + 0.153i)3-s + (0.123 + 0.214i)4-s + (−0.0654 + 0.997i)5-s + (−0.639 − 0.171i)6-s + (−0.803 − 1.39i)7-s − 0.840i·8-s + (−0.561 + 0.324i)9-s + (−0.620 + 0.928i)10-s + (0.472 − 0.126i)11-s + (−0.103 − 0.103i)12-s − 1.79i·14-s + (−0.115 − 0.581i)15-s + (0.593 − 1.02i)16-s + (0.388 − 1.45i)17-s − 0.724·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.438 + 0.898i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.438 + 0.898i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04592 - 0.653496i\)
\(L(\frac12)\) \(\approx\) \(1.04592 - 0.653496i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.146 - 2.23i)T \)
13 \( 1 \)
good2 \( 1 + (-1.36 - 0.789i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.991 - 0.265i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (2.12 + 3.68i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.56 + 0.419i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.60 + 5.98i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.100 + 0.374i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.600 + 2.24i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (2.42 + 1.39i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.30 + 2.30i)T - 31iT^{2} \)
37 \( 1 + (1.02 - 1.76i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.323 - 1.20i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.29 + 0.345i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + 0.483T + 47T^{2} \)
53 \( 1 + (7.24 + 7.24i)T + 53iT^{2} \)
59 \( 1 + (-0.425 - 0.114i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (5.64 + 9.77i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.16 - 3.55i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-10.1 - 2.71i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 - 4.96iT - 73T^{2} \)
79 \( 1 - 10.7iT - 79T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 + (-0.404 - 1.50i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-9.34 + 5.39i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.01390705081874295282823715674, −9.636405492905336737801693581571, −7.939182782525326411383574387167, −6.96021375447787526203443187797, −6.60643408183414132238018325188, −5.71578880168469657191182316562, −4.71584415276900295789706333777, −3.78511644235992881457284926108, −2.94606376076899098505552251674, −0.46682250064289049469495815831, 1.71409188285457500713449386410, 3.05407653110626538813154428203, 3.94368486836876784490138432899, 5.11906882185240344386226426893, 5.77110326484415012121519311444, 6.31327046344645953070299914712, 7.998704561076118230950548334131, 8.812752214454630134125887794528, 9.331947269670655971715044230918, 10.61259038635113443587068539270

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