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Properties

Label 2-1680-420.419-c0-0-12
Degree $2$
Conductor $1680$
Sign $i$
Analytic cond. $0.838429$
Root an. cond. $0.915657$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s − i·5-s + i·7-s − 9-s + 2·11-s − 15-s − 2i·17-s + 21-s − 25-s + i·27-s − 2i·33-s + 35-s + i·45-s − 49-s − 2·51-s + ⋯
L(s)  = 1  − i·3-s − i·5-s + i·7-s − 9-s + 2·11-s − 15-s − 2i·17-s + 21-s − 25-s + i·27-s − 2i·33-s + 35-s + i·45-s − 49-s − 2·51-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (1679, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :0),\ i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.171853790\)
\(L(\frac12)\) \(\approx\) \(1.171853790\)
\(L(1)\) 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 \( 1 + iT \)
5 \( 1 + iT \)
7 \( 1 - iT \)
good11 \( 1 - 2T + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + 2iT - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + 2T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.004259295231097020379755249834, −8.841075251881717388011387536461, −7.75850373884608958141230820740, −6.92381836269622197571106504118, −6.16281940535967806674520795658, −5.38052600781253761979822137407, −4.49298364270418865009472036733, −3.20201968297314379161139723705, −2.04716698239093996967656510522, −1.01801164324861966358658862034, 1.65659170779402261808463021296, 3.25607638351579060458976985379, 3.90231753188054546753137535744, 4.38711145040366245754871301621, 5.92310112654424420066303783948, 6.41604577779688997487994077624, 7.25913818085555142151326955995, 8.292556042668229174681115285068, 9.057527061169853624473468530073, 9.949151063465877945349328240979

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