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Properties

Label 2-1680-35.27-c1-0-21
Degree $2$
Conductor $1680$
Sign $0.685 - 0.727i$
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 + (−0.0188 + 2.23i)5-s + (2.30 + 1.30i)7-s + 1.00i·9-s + 5.19·11-s + (−0.817 − 0.817i)13-s + (1.59 − 1.56i)15-s + (−0.550 + 0.550i)17-s + 1.30·19-s + (−0.705 − 2.54i)21-s + (3.04 − 3.04i)23-s + (−4.99 − 0.0841i)25-s + (0.707 − 0.707i)27-s + 2.99i·29-s + 3.89i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.00841 + 0.999i)5-s + (0.870 + 0.492i)7-s + 0.333i·9-s + 1.56·11-s + (−0.226 − 0.226i)13-s + (0.411 − 0.404i)15-s + (−0.133 + 0.133i)17-s + 0.299·19-s + (−0.154 − 0.556i)21-s + (0.635 − 0.635i)23-s + (−0.999 − 0.0168i)25-s + (0.136 − 0.136i)27-s + 0.555i·29-s + 0.699i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.685 - 0.727i$
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: \((2,\ 1680,\ (\ :1/2),\ 0.685 - 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.768395320\)
\(L(\frac12)\) \(\approx\) \(1.768395320\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0188 - 2.23i)T \)
7 \( 1 + (-2.30 - 1.30i)T \)
good11 \( 1 - 5.19T + 11T^{2} \)
13 \( 1 + (0.817 + 0.817i)T + 13iT^{2} \)
17 \( 1 + (0.550 - 0.550i)T - 17iT^{2} \)
19 \( 1 - 1.30T + 19T^{2} \)
23 \( 1 + (-3.04 + 3.04i)T - 23iT^{2} \)
29 \( 1 - 2.99iT - 29T^{2} \)
31 \( 1 - 3.89iT - 31T^{2} \)
37 \( 1 + (-0.199 - 0.199i)T + 37iT^{2} \)
41 \( 1 + 2.72iT - 41T^{2} \)
43 \( 1 + (1.98 - 1.98i)T - 43iT^{2} \)
47 \( 1 + (-4.41 + 4.41i)T - 47iT^{2} \)
53 \( 1 + (1.09 - 1.09i)T - 53iT^{2} \)
59 \( 1 + 13.8T + 59T^{2} \)
61 \( 1 - 10.9iT - 61T^{2} \)
67 \( 1 + (-10.0 - 10.0i)T + 67iT^{2} \)
71 \( 1 - 16.2T + 71T^{2} \)
73 \( 1 + (5.32 + 5.32i)T + 73iT^{2} \)
79 \( 1 + 8.15iT - 79T^{2} \)
83 \( 1 + (7.49 + 7.49i)T + 83iT^{2} \)
89 \( 1 - 16.5T + 89T^{2} \)
97 \( 1 + (9.70 - 9.70i)T - 97iT^{2} \)
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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

−9.349572134826725694958339203287, −8.685699322919442537105936024907, −7.72812179207716623222453741375, −6.95371778850701645649887261468, −6.38401610779138764679911086615, −5.49294289274078571016518226576, −4.53487079869976636182713286017, −3.45212123040419574211585551670, −2.33544769873909304821057101170, −1.26644581447931516043963039575, 0.846308814619290447623079405706, 1.80146556619391817778123652531, 3.61206730883761348165392855071, 4.37647872842815565225445918140, 4.96293264330401488840914661020, 5.89644678681201144024851268243, 6.82331749061930618817351782269, 7.74132150643654524537934290132, 8.540104619526862158014481981551, 9.437232320501630532695691130870

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