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Properties

Label 2-1680-35.13-c1-0-35
Degree $2$
Conductor $1680$
Sign $0.0240 + 0.999i$
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 + (1.75 − 1.38i)5-s + (−2.02 + 1.69i)7-s − 1.00i·9-s − 2.43·11-s + (1.08 − 1.08i)13-s + (0.265 − 2.22i)15-s + (−1.74 − 1.74i)17-s + 6.07·19-s + (−0.234 + 2.63i)21-s + (1.45 + 1.45i)23-s + (1.17 − 4.85i)25-s + (−0.707 − 0.707i)27-s − 5.62i·29-s − 6.36i·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.786 − 0.618i)5-s + (−0.766 + 0.641i)7-s − 0.333i·9-s − 0.732·11-s + (0.300 − 0.300i)13-s + (0.0685 − 0.573i)15-s + (−0.422 − 0.422i)17-s + 1.39·19-s + (−0.0511 + 0.575i)21-s + (0.303 + 0.303i)23-s + (0.235 − 0.971i)25-s + (−0.136 − 0.136i)27-s − 1.04i·29-s − 1.14i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0240 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0240 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.0240 + 0.999i$
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: \((2,\ 1680,\ (\ :1/2),\ 0.0240 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.870088363\)
\(L(\frac12)\) \(\approx\) \(1.870088363\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.75 + 1.38i)T \)
7 \( 1 + (2.02 - 1.69i)T \)
good11 \( 1 + 2.43T + 11T^{2} \)
13 \( 1 + (-1.08 + 1.08i)T - 13iT^{2} \)
17 \( 1 + (1.74 + 1.74i)T + 17iT^{2} \)
19 \( 1 - 6.07T + 19T^{2} \)
23 \( 1 + (-1.45 - 1.45i)T + 23iT^{2} \)
29 \( 1 + 5.62iT - 29T^{2} \)
31 \( 1 + 6.36iT - 31T^{2} \)
37 \( 1 + (-6.09 + 6.09i)T - 37iT^{2} \)
41 \( 1 + 7.22iT - 41T^{2} \)
43 \( 1 + (6.91 + 6.91i)T + 43iT^{2} \)
47 \( 1 + (1.47 + 1.47i)T + 47iT^{2} \)
53 \( 1 + (-6.45 - 6.45i)T + 53iT^{2} \)
59 \( 1 - 3.00T + 59T^{2} \)
61 \( 1 - 9.57iT - 61T^{2} \)
67 \( 1 + (-4.18 + 4.18i)T - 67iT^{2} \)
71 \( 1 + 1.97T + 71T^{2} \)
73 \( 1 + (0.625 - 0.625i)T - 73iT^{2} \)
79 \( 1 + 0.692iT - 79T^{2} \)
83 \( 1 + (12.1 - 12.1i)T - 83iT^{2} \)
89 \( 1 + 6.14T + 89T^{2} \)
97 \( 1 + (-9.98 - 9.98i)T + 97iT^{2} \)
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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

−9.259011549467089761350502292657, −8.456409440945717567071463878953, −7.62523192625837593736840564415, −6.76695628291108775065613029158, −5.67641436127385903175492579253, −5.44771726494578788428211866407, −4.03936616023053413565857522538, −2.84152134657133525249306468463, −2.17896651123213743025216040258, −0.68496656362758011734969641664, 1.45793420826083313310772571014, 2.91151704832362531181856277864, 3.29766474819217243008417609320, 4.58588807821909737997312359087, 5.45437923154840499549636018486, 6.49162494348440639725503330847, 7.01332809094736268604762505473, 7.992017653219504980633179988296, 8.885310320306110770927909926089, 9.807094383340841489867470795555

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