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Properties

Label 2-833-17.4-c1-0-41
Degree $2$
Conductor $833$
Sign $0.908 - 0.418i$
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  + 2.46i·2-s + (1.82 − 1.82i)3-s − 4.05·4-s + (2.12 − 2.12i)5-s + (4.50 + 4.50i)6-s − 5.04i·8-s − 3.69i·9-s + (5.22 + 5.22i)10-s + (0.0608 + 0.0608i)11-s + (−7.41 + 7.41i)12-s + 4.28·13-s − 7.77i·15-s + 4.31·16-s + (−4.01 − 0.946i)17-s + 9.08·18-s − 6.85i·19-s + ⋯
L(s)  = 1  + 1.73i·2-s + (1.05 − 1.05i)3-s − 2.02·4-s + (0.949 − 0.949i)5-s + (1.83 + 1.83i)6-s − 1.78i·8-s − 1.23i·9-s + (1.65 + 1.65i)10-s + (0.0183 + 0.0183i)11-s + (−2.14 + 2.14i)12-s + 1.18·13-s − 2.00i·15-s + 1.07·16-s + (−0.973 − 0.229i)17-s + 2.14·18-s − 1.57i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(833\)    =    \(7^{2} \cdot 17\)
Sign: $0.908 - 0.418i$
Analytic conductor: \(6.65153\)
Root analytic conductor: \(2.57905\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{833} (344, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 833,\ (\ :1/2),\ 0.908 - 0.418i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.25657 + 0.494343i\)
\(L(\frac12)\) \(\approx\) \(2.25657 + 0.494343i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.01 + 0.946i)T \)
good2 \( 1 - 2.46iT - 2T^{2} \)
3 \( 1 + (-1.82 + 1.82i)T - 3iT^{2} \)
5 \( 1 + (-2.12 + 2.12i)T - 5iT^{2} \)
11 \( 1 + (-0.0608 - 0.0608i)T + 11iT^{2} \)
13 \( 1 - 4.28T + 13T^{2} \)
19 \( 1 + 6.85iT - 19T^{2} \)
23 \( 1 + (0.498 + 0.498i)T + 23iT^{2} \)
29 \( 1 + (-1.07 + 1.07i)T - 29iT^{2} \)
31 \( 1 + (0.571 - 0.571i)T - 31iT^{2} \)
37 \( 1 + (2.72 - 2.72i)T - 37iT^{2} \)
41 \( 1 + (-0.0255 - 0.0255i)T + 41iT^{2} \)
43 \( 1 - 8.49iT - 43T^{2} \)
47 \( 1 - 9.76T + 47T^{2} \)
53 \( 1 - 11.8iT - 53T^{2} \)
59 \( 1 - 4.25iT - 59T^{2} \)
61 \( 1 + (-1.78 - 1.78i)T + 61iT^{2} \)
67 \( 1 - 4.92T + 67T^{2} \)
71 \( 1 + (-4.95 + 4.95i)T - 71iT^{2} \)
73 \( 1 + (2.69 - 2.69i)T - 73iT^{2} \)
79 \( 1 + (-4.64 - 4.64i)T + 79iT^{2} \)
83 \( 1 + 3.86iT - 83T^{2} \)
89 \( 1 + 5.04T + 89T^{2} \)
97 \( 1 + (6.55 - 6.55i)T - 97iT^{2} \)
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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

−9.503543668634478188758701094166, −8.866984645798793060534630179284, −8.614310419377456573518010849284, −7.67785858184199607998785046506, −6.81421927941168699692626005404, −6.22998299685009073937742119085, −5.25003899813043865950571020031, −4.28341809279795852944939566034, −2.55775425934985275685964839392, −1.13323136077282880706368964367, 1.80640110050506561268876321674, 2.57242874172781521935859790049, 3.63552437723212534537332753197, 3.96318836125742311692959986021, 5.39917727152312815381275433224, 6.57064185205794652129280016976, 8.227804144808133683171690855607, 8.916500356930638211785263624801, 9.591443754522218167331785752518, 10.39991173870155817679802112521

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