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Properties

Label 2-1680-35.13-c1-0-39
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: $\chi_{1680} (433, \cdot )$
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

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

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