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Properties

Label 2-1740-145.128-c1-0-11
Degree $2$
Conductor $1740$
Sign $-0.0814 - 0.996i$
Analytic cond. $13.8939$
Root an. cond. $3.72746$
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  + i·3-s + (1.73 + 1.40i)5-s + (−0.403 + 0.403i)7-s − 9-s + (−1.63 − 1.63i)11-s + (1.65 − 1.65i)13-s + (−1.40 + 1.73i)15-s + 1.73·17-s + (−0.837 + 0.837i)19-s + (−0.403 − 0.403i)21-s + (5.89 + 5.89i)23-s + (1.04 + 4.88i)25-s − i·27-s + (−2.32 + 4.85i)29-s + (5.79 + 5.79i)31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.777 + 0.628i)5-s + (−0.152 + 0.152i)7-s − 0.333·9-s + (−0.494 − 0.494i)11-s + (0.460 − 0.460i)13-s + (−0.363 + 0.448i)15-s + 0.421·17-s + (−0.192 + 0.192i)19-s + (−0.0880 − 0.0880i)21-s + (1.22 + 1.22i)23-s + (0.209 + 0.977i)25-s − 0.192i·27-s + (−0.431 + 0.902i)29-s + (1.04 + 1.04i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0814 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0814 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1740\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 29\)
Sign: $-0.0814 - 0.996i$
Analytic conductor: \(13.8939\)
Root analytic conductor: \(3.72746\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1740} (853, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1740,\ (\ :1/2),\ -0.0814 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.824817425\)
\(L(\frac12)\) \(\approx\) \(1.824817425\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 + (-1.73 - 1.40i)T \)
29 \( 1 + (2.32 - 4.85i)T \)
good7 \( 1 + (0.403 - 0.403i)T - 7iT^{2} \)
11 \( 1 + (1.63 + 1.63i)T + 11iT^{2} \)
13 \( 1 + (-1.65 + 1.65i)T - 13iT^{2} \)
17 \( 1 - 1.73T + 17T^{2} \)
19 \( 1 + (0.837 - 0.837i)T - 19iT^{2} \)
23 \( 1 + (-5.89 - 5.89i)T + 23iT^{2} \)
31 \( 1 + (-5.79 - 5.79i)T + 31iT^{2} \)
37 \( 1 + 8.68iT - 37T^{2} \)
41 \( 1 + (-1.82 + 1.82i)T - 41iT^{2} \)
43 \( 1 - 10.5iT - 43T^{2} \)
47 \( 1 - 10.9iT - 47T^{2} \)
53 \( 1 + (2.48 + 2.48i)T + 53iT^{2} \)
59 \( 1 + 3.22iT - 59T^{2} \)
61 \( 1 + (4.86 + 4.86i)T + 61iT^{2} \)
67 \( 1 + (4.86 + 4.86i)T + 67iT^{2} \)
71 \( 1 - 4.15iT - 71T^{2} \)
73 \( 1 + 7.08T + 73T^{2} \)
79 \( 1 + (7.63 - 7.63i)T - 79iT^{2} \)
83 \( 1 + (5.19 + 5.19i)T + 83iT^{2} \)
89 \( 1 + (-0.214 + 0.214i)T - 89iT^{2} \)
97 \( 1 + 16.9iT - 97T^{2} \)
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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

−9.486603404700671643355170201146, −8.975448418686144107692563674144, −7.954005121929019298452638075584, −7.13606620747649710563564489807, −6.06485522326948867853611433990, −5.61086912252962367396923511428, −4.69028049668917381104695750350, −3.27675532726047909902717544554, −2.94419331960972607744833968004, −1.39902363684049590624318086796, 0.72425348202044532881110409149, 1.93087836086581118309534084430, 2.81843799136897991039993522820, 4.25042016020027829422683883582, 5.04936651517785641287184527366, 5.97424855720617308153727426589, 6.64813435632539082884794096372, 7.48090797508362998763988600882, 8.470289872605415570936818607922, 8.927255054550598600011052276647

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