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Properties

Label 2-1680-35.27-c1-0-7
Degree $2$
Conductor $1680$
Sign $-0.582 - 0.812i$
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 + (−1.98 − 1.02i)5-s + (2.56 + 0.630i)7-s + 1.00i·9-s − 2.36·11-s + (−0.918 − 0.918i)13-s + (−0.679 − 2.13i)15-s + (−3.62 + 3.62i)17-s − 1.07·19-s + (1.37 + 2.26i)21-s + (−1.45 + 1.45i)23-s + (2.89 + 4.07i)25-s + (−0.707 + 0.707i)27-s + 7.72i·29-s − 1.21i·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−0.888 − 0.458i)5-s + (0.971 + 0.238i)7-s + 0.333i·9-s − 0.712·11-s + (−0.254 − 0.254i)13-s + (−0.175 − 0.550i)15-s + (−0.878 + 0.878i)17-s − 0.247·19-s + (0.299 + 0.493i)21-s + (−0.302 + 0.302i)23-s + (0.578 + 0.815i)25-s + (−0.136 + 0.136i)27-s + 1.43i·29-s − 0.218i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.582 - 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.582 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.582 - 0.812i$
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: \((2,\ 1680,\ (\ :1/2),\ -0.582 - 0.812i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.009017855\)
\(L(\frac12)\) \(\approx\) \(1.009017855\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.98 + 1.02i)T \)
7 \( 1 + (-2.56 - 0.630i)T \)
good11 \( 1 + 2.36T + 11T^{2} \)
13 \( 1 + (0.918 + 0.918i)T + 13iT^{2} \)
17 \( 1 + (3.62 - 3.62i)T - 17iT^{2} \)
19 \( 1 + 1.07T + 19T^{2} \)
23 \( 1 + (1.45 - 1.45i)T - 23iT^{2} \)
29 \( 1 - 7.72iT - 29T^{2} \)
31 \( 1 + 1.21iT - 31T^{2} \)
37 \( 1 + (-2.38 - 2.38i)T + 37iT^{2} \)
41 \( 1 - 8.42iT - 41T^{2} \)
43 \( 1 + (0.879 - 0.879i)T - 43iT^{2} \)
47 \( 1 + (-3.67 + 3.67i)T - 47iT^{2} \)
53 \( 1 + (7.88 - 7.88i)T - 53iT^{2} \)
59 \( 1 + 8.24T + 59T^{2} \)
61 \( 1 + 10.1iT - 61T^{2} \)
67 \( 1 + (6.62 + 6.62i)T + 67iT^{2} \)
71 \( 1 - 8.21T + 71T^{2} \)
73 \( 1 + (1.79 + 1.79i)T + 73iT^{2} \)
79 \( 1 - 12.2iT - 79T^{2} \)
83 \( 1 + (0.387 + 0.387i)T + 83iT^{2} \)
89 \( 1 + 7.46T + 89T^{2} \)
97 \( 1 + (6.92 - 6.92i)T - 97iT^{2} \)
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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

−9.427430030387916058075286449909, −8.731945351943641487952298411982, −8.009320435072117412448012716580, −7.68502220831048525398910470296, −6.43796649943431720072188705526, −5.21283475612040625769589165685, −4.69629534550544070981349661708, −3.84760891598633652847072843046, −2.76313206029139500201972270850, −1.54024465180617954258145342262, 0.35480747828321573847097664531, 2.05269814989151062982542329620, 2.86550326325037397238338333406, 4.13568398066485292482885435819, 4.69025105652311475743680721084, 5.90384367123806379434753997838, 7.02873427449094698048574894078, 7.47295939127852148903224703462, 8.198083444599979232897794516683, 8.813748979801087587829589465605

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