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Properties

Label 2-845-65.63-c1-0-39
Degree $2$
Conductor $845$
Sign $0.917 - 0.396i$
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  + (0.5 + 0.866i)2-s + (0.366 − 1.36i)3-s + (0.500 − 0.866i)4-s + (1 + 2i)5-s + (1.36 − 0.366i)6-s + (1.73 + i)7-s + 3·8-s + (0.866 + 0.5i)9-s + (−1.23 + 1.86i)10-s + (−1.36 − 0.366i)11-s + (−0.999 − i)12-s + 1.99i·14-s + (3.09 − 0.633i)15-s + (0.500 + 0.866i)16-s + (−1.36 + 0.366i)17-s + i·18-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.211 − 0.788i)3-s + (0.250 − 0.433i)4-s + (0.447 + 0.894i)5-s + (0.557 − 0.149i)6-s + (0.654 + 0.377i)7-s + 1.06·8-s + (0.288 + 0.166i)9-s + (−0.389 + 0.590i)10-s + (−0.411 − 0.110i)11-s + (−0.288 − 0.288i)12-s + 0.534i·14-s + (0.799 − 0.163i)15-s + (0.125 + 0.216i)16-s + (−0.331 + 0.0887i)17-s + 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.396i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 - 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.917 - 0.396i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (258, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.917 - 0.396i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.59886 + 0.537487i\)
\(L(\frac12)\) \(\approx\) \(2.59886 + 0.537487i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1 - 2i)T \)
13 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.366 + 1.36i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-1.73 - i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.36 + 0.366i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.36 - 0.366i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.83 - 6.83i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (4.09 + 1.09i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5 + 5i)T + 31iT^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.56 + 9.56i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.366 + 1.36i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 + (-5 - 5i)T + 53iT^{2} \)
59 \( 1 + (9.56 - 2.56i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-7 + 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.36 + 0.366i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 - 2iT - 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + (1.83 - 6.83i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1 + 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.40089841420372590181250981291, −9.493953172539956142302085520266, −8.010864123378539633833889195881, −7.66028139637676462348798495840, −6.75086653022207688765271736544, −5.96820472201886799103432174863, −5.31050350857847089071952504598, −3.97225444604145954228191447332, −2.31472829870757840563730726406, −1.68195121795393751724503552857, 1.39015827234303536411125646666, 2.63479667694334359644118640588, 3.85707514246075955282663668116, 4.62206790813101538484373944832, 5.20254827087956387695690313977, 6.77606438385546331261354019385, 7.70064642317251386226541051755, 8.597876888699161899544384827170, 9.425434684964134164409379871918, 10.20764877718693279213756394878

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