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Properties

Label 2-845-65.47-c1-0-21
Degree $2$
Conductor $845$
Sign $0.383 - 0.923i$
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  + 0.131i·2-s + (−0.243 + 0.243i)3-s + 1.98·4-s + (−2.08 − 0.813i)5-s + (−0.0319 − 0.0319i)6-s + 2.78·7-s + 0.522i·8-s + 2.88i·9-s + (0.106 − 0.273i)10-s + (−2.86 + 2.86i)11-s + (−0.482 + 0.482i)12-s + 0.365i·14-s + (0.704 − 0.308i)15-s + 3.89·16-s + (−1.71 + 1.71i)17-s − 0.378·18-s + ⋯
L(s)  = 1  + 0.0928i·2-s + (−0.140 + 0.140i)3-s + 0.991·4-s + (−0.931 − 0.363i)5-s + (−0.0130 − 0.0130i)6-s + 1.05·7-s + 0.184i·8-s + 0.960i·9-s + (0.0337 − 0.0864i)10-s + (−0.864 + 0.864i)11-s + (−0.139 + 0.139i)12-s + 0.0976i·14-s + (0.181 − 0.0797i)15-s + 0.974·16-s + (−0.415 + 0.415i)17-s − 0.0891·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.383 - 0.923i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.383 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34481 + 0.897649i\)
\(L(\frac12)\) \(\approx\) \(1.34481 + 0.897649i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.08 + 0.813i)T \)
13 \( 1 \)
good2 \( 1 - 0.131iT - 2T^{2} \)
3 \( 1 + (0.243 - 0.243i)T - 3iT^{2} \)
7 \( 1 - 2.78T + 7T^{2} \)
11 \( 1 + (2.86 - 2.86i)T - 11iT^{2} \)
17 \( 1 + (1.71 - 1.71i)T - 17iT^{2} \)
19 \( 1 + (1.34 - 1.34i)T - 19iT^{2} \)
23 \( 1 + (-5.64 - 5.64i)T + 23iT^{2} \)
29 \( 1 + 4.57iT - 29T^{2} \)
31 \( 1 + (3.87 + 3.87i)T + 31iT^{2} \)
37 \( 1 - 7.01T + 37T^{2} \)
41 \( 1 + (-4.54 - 4.54i)T + 41iT^{2} \)
43 \( 1 + (-4.57 - 4.57i)T + 43iT^{2} \)
47 \( 1 - 0.512T + 47T^{2} \)
53 \( 1 + (1.32 - 1.32i)T - 53iT^{2} \)
59 \( 1 + (-1.85 - 1.85i)T + 59iT^{2} \)
61 \( 1 + 1.28T + 61T^{2} \)
67 \( 1 + 3.61iT - 67T^{2} \)
71 \( 1 + (4.54 + 4.54i)T + 71iT^{2} \)
73 \( 1 - 9.93iT - 73T^{2} \)
79 \( 1 + 8.37iT - 79T^{2} \)
83 \( 1 + 3.17T + 83T^{2} \)
89 \( 1 + (4.40 + 4.40i)T + 89iT^{2} \)
97 \( 1 - 11.7iT - 97T^{2} \)
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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

−10.71879285173691428314109314364, −9.591797208353423134906707584190, −8.273448355347056787999059571557, −7.70328811411742297133289807586, −7.32449758366342254257227136998, −5.87312361146268882647081836295, −4.96339269911374488579681077609, −4.22850449951973310771153010657, −2.67879317249199878433806318932, −1.61692529841535266667236747447, 0.824574418863889243187708578200, 2.51483757303268521519839197403, 3.38474056406518403078066307100, 4.62299766618506117821146338125, 5.73469106416047515754200568845, 6.80127565349326840929840607186, 7.33218642810825614187829507748, 8.271637686287766976609188251552, 8.967666307238778093806198048885, 10.49546598175237808356316417431

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