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Properties

Label 2-833-17.9-c1-0-12
Degree $2$
Conductor $833$
Sign $0.427 - 0.904i$
Analytic cond. $6.65153$
Root an. cond. $2.57905$
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  + (0.309 − 0.309i)2-s + (−2.18 − 0.905i)3-s + 1.80i·4-s + (0.192 − 0.465i)5-s + (−0.955 + 0.395i)6-s + (1.17 + 1.17i)8-s + (1.83 + 1.83i)9-s + (−0.0842 − 0.203i)10-s + (2.37 − 0.983i)11-s + (1.63 − 3.95i)12-s + 0.364i·13-s + (−0.842 + 0.842i)15-s − 2.88·16-s + (−3.64 − 1.93i)17-s + 1.13·18-s + (−1.60 + 1.60i)19-s + ⋯
L(s)  = 1  + (0.218 − 0.218i)2-s + (−1.26 − 0.522i)3-s + 0.904i·4-s + (0.0861 − 0.208i)5-s + (−0.390 + 0.161i)6-s + (0.416 + 0.416i)8-s + (0.611 + 0.611i)9-s + (−0.0266 − 0.0643i)10-s + (0.716 − 0.296i)11-s + (0.472 − 1.14i)12-s + 0.101i·13-s + (−0.217 + 0.217i)15-s − 0.722·16-s + (−0.883 − 0.469i)17-s + 0.267·18-s + (−0.368 + 0.368i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.427 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.427 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(833\)    =    \(7^{2} \cdot 17\)
Sign: $0.427 - 0.904i$
Analytic conductor: \(6.65153\)
Root analytic conductor: \(2.57905\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{833} (638, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 833,\ (\ :1/2),\ 0.427 - 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.753988 + 0.477492i\)
\(L(\frac12)\) \(\approx\) \(0.753988 + 0.477492i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
17 \( 1 + (3.64 + 1.93i)T \)
good2 \( 1 + (-0.309 + 0.309i)T - 2iT^{2} \)
3 \( 1 + (2.18 + 0.905i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (-0.192 + 0.465i)T + (-3.53 - 3.53i)T^{2} \)
11 \( 1 + (-2.37 + 0.983i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 - 0.364iT - 13T^{2} \)
19 \( 1 + (1.60 - 1.60i)T - 19iT^{2} \)
23 \( 1 + (4.60 - 1.90i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (1.22 - 2.96i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (-8.10 - 3.35i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (-9.09 - 3.76i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-3.41 - 8.24i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (0.580 + 0.580i)T + 43iT^{2} \)
47 \( 1 - 0.763iT - 47T^{2} \)
53 \( 1 + (3.95 - 3.95i)T - 53iT^{2} \)
59 \( 1 + (0.396 + 0.396i)T + 59iT^{2} \)
61 \( 1 + (-1.68 - 4.07i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 5.80T + 67T^{2} \)
71 \( 1 + (7.04 + 2.91i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (0.0934 - 0.225i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-10.4 + 4.31i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (11.8 - 11.8i)T - 83iT^{2} \)
89 \( 1 - 15.9iT - 89T^{2} \)
97 \( 1 + (2.45 - 5.91i)T + (-68.5 - 68.5i)T^{2} \)
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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

−10.76984796615293271403236641293, −9.510395813990489686497097201385, −8.606663103675507129635796967273, −7.69590225616156992770994454415, −6.71679949167356916935852721937, −6.15821653378622722559376126151, −4.96475883798108813881220306579, −4.17425752166654348235260950624, −2.85622306001289125684441503023, −1.32339824846691596885103573158, 0.52191983194133588849418617814, 2.20968392564761614687828609556, 4.30720896186554258185753330020, 4.56863371728427805988241767702, 5.94852703242623433054557651866, 6.16125578760568706831552649944, 7.05723778726591253646518055481, 8.467659547473058016010302750833, 9.574787050782210899485170047549, 10.18087984418146733431939758311

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