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Properties

Label 2-1680-35.27-c1-0-39
Degree $2$
Conductor $1680$
Sign $0.968 - 0.248i$
Analytic cond. $13.4148$
Root an. cond. $3.66263$
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 + (2.17 − 0.513i)5-s + (2.56 − 0.659i)7-s + 1.00i·9-s + 1.57·11-s + (3.14 + 3.14i)13-s + (1.90 + 1.17i)15-s + (4.47 − 4.47i)17-s − 6.38·19-s + (2.27 + 1.34i)21-s + (−1.38 + 1.38i)23-s + (4.47 − 2.23i)25-s + (−0.707 + 0.707i)27-s + 2.19i·29-s + 1.53i·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.973 − 0.229i)5-s + (0.968 − 0.249i)7-s + 0.333i·9-s + 0.475·11-s + (0.872 + 0.872i)13-s + (0.491 + 0.303i)15-s + (1.08 − 1.08i)17-s − 1.46·19-s + (0.497 + 0.293i)21-s + (−0.288 + 0.288i)23-s + (0.894 − 0.446i)25-s + (−0.136 + 0.136i)27-s + 0.407i·29-s + 0.276i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.968 - 0.248i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ 0.968 - 0.248i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.835878087\)
\(L(\frac12)\) \(\approx\) \(2.835878087\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.17 + 0.513i)T \)
7 \( 1 + (-2.56 + 0.659i)T \)
good11 \( 1 - 1.57T + 11T^{2} \)
13 \( 1 + (-3.14 - 3.14i)T + 13iT^{2} \)
17 \( 1 + (-4.47 + 4.47i)T - 17iT^{2} \)
19 \( 1 + 6.38T + 19T^{2} \)
23 \( 1 + (1.38 - 1.38i)T - 23iT^{2} \)
29 \( 1 - 2.19iT - 29T^{2} \)
31 \( 1 - 1.53iT - 31T^{2} \)
37 \( 1 + (8.06 + 8.06i)T + 37iT^{2} \)
41 \( 1 - 4.79iT - 41T^{2} \)
43 \( 1 + (0.0831 - 0.0831i)T - 43iT^{2} \)
47 \( 1 + (3.14 - 3.14i)T - 47iT^{2} \)
53 \( 1 + (-6.30 + 6.30i)T - 53iT^{2} \)
59 \( 1 + 7.59T + 59T^{2} \)
61 \( 1 - 3.73iT - 61T^{2} \)
67 \( 1 + (-1.84 - 1.84i)T + 67iT^{2} \)
71 \( 1 + 9.63T + 71T^{2} \)
73 \( 1 + (2.46 + 2.46i)T + 73iT^{2} \)
79 \( 1 + 15.8iT - 79T^{2} \)
83 \( 1 + (-2.70 - 2.70i)T + 83iT^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 + (8.82 - 8.82i)T - 97iT^{2} \)
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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

−9.126687549600215385547986069086, −8.886233653131601008802792473701, −7.941223186552649084604200056169, −6.98413218955551107814137207203, −6.09910174931449825087640357998, −5.18898007838566409000526686739, −4.44541815144818426165924654808, −3.50529315893415092375118511487, −2.16572323317162760163047769209, −1.35324979983607860437389285589, 1.32933590231546861851004115367, 2.04886421440240392488776550975, 3.21574592953575535900819533494, 4.24788135328921680705592394488, 5.48232955065775279954576404431, 6.05231430797703644502647965744, 6.82745444858407334856042381500, 8.014713456896656052208607199453, 8.418073125324602247178904230132, 9.131509749796277741059509840219

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