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Properties

Label 2-840-35.13-c1-0-2
Degree $2$
Conductor $840$
Sign $-0.776 - 0.629i$
Analytic cond. $6.70743$
Root an. cond. $2.58987$
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  + (0.707 − 0.707i)3-s + (0.503 + 2.17i)5-s + (−2.48 − 0.918i)7-s − 1.00i·9-s − 5.08·11-s + (−3.79 + 3.79i)13-s + (1.89 + 1.18i)15-s + (2.83 + 2.83i)17-s − 3.66·19-s + (−2.40 + 1.10i)21-s + (−0.591 − 0.591i)23-s + (−4.49 + 2.19i)25-s + (−0.707 − 0.707i)27-s − 1.05i·29-s + 8.77i·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.225 + 0.974i)5-s + (−0.937 − 0.347i)7-s − 0.333i·9-s − 1.53·11-s + (−1.05 + 1.05i)13-s + (0.489 + 0.305i)15-s + (0.687 + 0.687i)17-s − 0.840·19-s + (−0.524 + 0.241i)21-s + (−0.123 − 0.123i)23-s + (−0.898 + 0.438i)25-s + (−0.136 − 0.136i)27-s − 0.195i·29-s + 1.57i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 - 0.629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.776 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.776 - 0.629i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.776 - 0.629i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.188206 + 0.531088i\)
\(L(\frac12)\) \(\approx\) \(0.188206 + 0.531088i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-0.503 - 2.17i)T \)
7 \( 1 + (2.48 + 0.918i)T \)
good11 \( 1 + 5.08T + 11T^{2} \)
13 \( 1 + (3.79 - 3.79i)T - 13iT^{2} \)
17 \( 1 + (-2.83 - 2.83i)T + 17iT^{2} \)
19 \( 1 + 3.66T + 19T^{2} \)
23 \( 1 + (0.591 + 0.591i)T + 23iT^{2} \)
29 \( 1 + 1.05iT - 29T^{2} \)
31 \( 1 - 8.77iT - 31T^{2} \)
37 \( 1 + (-4.95 + 4.95i)T - 37iT^{2} \)
41 \( 1 + 2.91iT - 41T^{2} \)
43 \( 1 + (6.86 + 6.86i)T + 43iT^{2} \)
47 \( 1 + (-1.97 - 1.97i)T + 47iT^{2} \)
53 \( 1 + (5.54 + 5.54i)T + 53iT^{2} \)
59 \( 1 + 1.49T + 59T^{2} \)
61 \( 1 - 12.3iT - 61T^{2} \)
67 \( 1 + (-4.30 + 4.30i)T - 67iT^{2} \)
71 \( 1 - 9.87T + 71T^{2} \)
73 \( 1 + (7.02 - 7.02i)T - 73iT^{2} \)
79 \( 1 - 11.9iT - 79T^{2} \)
83 \( 1 + (5.33 - 5.33i)T - 83iT^{2} \)
89 \( 1 + 1.75T + 89T^{2} \)
97 \( 1 + (-3.26 - 3.26i)T + 97iT^{2} \)
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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

−10.24803780007533161373843249459, −9.984832540995642433911414535498, −8.843395541361902733977454670450, −7.79448657091476101930851292869, −7.08633269287308215743510570505, −6.44923489697935760382691709398, −5.38042664136119535340397852154, −3.97503939037825785198477967139, −2.92095420617070287022002545573, −2.12643233281555177986961235280, 0.23459378978185167602925388367, 2.38800374267103693694772088619, 3.16908672366710518828088817050, 4.64380615277608990859693794433, 5.29791769153770965048069638256, 6.16643111892634740807862211395, 7.71298924566262366188059377647, 8.059734627828142848500159460511, 9.203500510409376684716258747739, 9.879112145120269627478433451965

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