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Properties

Label 2-845-65.49-c1-0-66
Degree $2$
Conductor $845$
Sign $-0.946 + 0.322i$
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 + (1.73 − i)3-s + (0.500 − 0.866i)4-s + (1 − 2i)5-s + (−1.73 − 0.999i)6-s − 3·8-s + (0.499 − 0.866i)9-s + (−2.23 + 0.133i)10-s + (−1.73 + i)11-s − 2i·12-s + (−0.267 − 4.46i)15-s + (0.500 + 0.866i)16-s − 0.999·18-s + (−5.19 − 3i)19-s + (−1.23 − 1.86i)20-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.999 − 0.577i)3-s + (0.250 − 0.433i)4-s + (0.447 − 0.894i)5-s + (−0.707 − 0.408i)6-s − 1.06·8-s + (0.166 − 0.288i)9-s + (−0.705 + 0.0423i)10-s + (−0.522 + 0.301i)11-s − 0.577i·12-s + (−0.0691 − 1.15i)15-s + (0.125 + 0.216i)16-s − 0.235·18-s + (−1.19 − 0.688i)19-s + (−0.275 − 0.417i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 + 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.946 + 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.946 + 0.322i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.946 + 0.322i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.291378 - 1.75985i\)
\(L(\frac12)\) \(\approx\) \(0.291378 - 1.75985i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 + i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.73 - i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.19 + 3i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.19 + 3i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6iT - 31T^{2} \)
37 \( 1 + (-3 - 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.92 + 4i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.19 - 3i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 + (-1.73 - i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6 - 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.73 + i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (-6.92 + 4i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (3 - 5.19i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.630148919357295903796747715913, −9.042671046391237485298566426427, −8.384657046438564940439044275735, −7.46328712647682042883918702954, −6.36361869385154719666093344303, −5.42438921614230462569667443482, −4.27763516946807894295886667467, −2.58101112202255311625536397743, −2.20656430170234767083132605709, −0.833855255442132338311186781335, 2.33237208351608781365439211813, 3.12645950263944234731331783311, 3.91459477670784607709572181358, 5.54198415126812038577382896140, 6.42063523458295172276351804896, 7.32491708798460488516764507324, 8.008726506361127103272203694981, 8.972489290916444059931481310142, 9.339780556030029524599873112163, 10.56104428480064941632511142117

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