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Properties

Label 2-1740-145.17-c1-0-12
Degree $2$
Conductor $1740$
Sign $0.695 + 0.718i$
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 + (−1.70 + 1.44i)5-s + (−0.746 − 0.746i)7-s − 9-s + (−1.32 + 1.32i)11-s + (1.13 + 1.13i)13-s + (1.44 + 1.70i)15-s + 0.116·17-s + (−1.77 − 1.77i)19-s + (−0.746 + 0.746i)21-s + (3.17 − 3.17i)23-s + (0.844 − 4.92i)25-s + i·27-s + (5.15 + 1.56i)29-s + (0.474 − 0.474i)31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.764 + 0.644i)5-s + (−0.282 − 0.282i)7-s − 0.333·9-s + (−0.398 + 0.398i)11-s + (0.314 + 0.314i)13-s + (0.372 + 0.441i)15-s + 0.0281·17-s + (−0.406 − 0.406i)19-s + (−0.162 + 0.162i)21-s + (0.661 − 0.661i)23-s + (0.168 − 0.985i)25-s + 0.192i·27-s + (0.956 + 0.290i)29-s + (0.0852 − 0.0852i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1740\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 29\)
Sign: $0.695 + 0.718i$
Analytic conductor: \(13.8939\)
Root analytic conductor: \(3.72746\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1740} (1177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1740,\ (\ :1/2),\ 0.695 + 0.718i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.230524984\)
\(L(\frac12)\) \(\approx\) \(1.230524984\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 - 1.44i)T \)
29 \( 1 + (-5.15 - 1.56i)T \)
good7 \( 1 + (0.746 + 0.746i)T + 7iT^{2} \)
11 \( 1 + (1.32 - 1.32i)T - 11iT^{2} \)
13 \( 1 + (-1.13 - 1.13i)T + 13iT^{2} \)
17 \( 1 - 0.116T + 17T^{2} \)
19 \( 1 + (1.77 + 1.77i)T + 19iT^{2} \)
23 \( 1 + (-3.17 + 3.17i)T - 23iT^{2} \)
31 \( 1 + (-0.474 + 0.474i)T - 31iT^{2} \)
37 \( 1 + 3.66iT - 37T^{2} \)
41 \( 1 + (-9.00 - 9.00i)T + 41iT^{2} \)
43 \( 1 + 1.16iT - 43T^{2} \)
47 \( 1 - 0.280iT - 47T^{2} \)
53 \( 1 + (-3.72 + 3.72i)T - 53iT^{2} \)
59 \( 1 + 3.01iT - 59T^{2} \)
61 \( 1 + (-0.759 + 0.759i)T - 61iT^{2} \)
67 \( 1 + (-8.81 + 8.81i)T - 67iT^{2} \)
71 \( 1 + 12.5iT - 71T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
79 \( 1 + (0.399 + 0.399i)T + 79iT^{2} \)
83 \( 1 + (4.85 - 4.85i)T - 83iT^{2} \)
89 \( 1 + (-6.10 - 6.10i)T + 89iT^{2} \)
97 \( 1 + 3.53iT - 97T^{2} \)
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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

−9.111224478976786205028276959333, −8.234810108784718463343327920322, −7.61942537093169110969694297527, −6.74436791094891939375411158602, −6.40316394844072292503223748056, −5.04641691101308578500279157628, −4.15038207213636751545028957994, −3.14123764110734236483290638252, −2.25357673139032796901909564848, −0.63124038248216795944438207979, 0.911817373995546591567019969943, 2.65607813052561634944779510358, 3.60763153142413635200664429419, 4.38716206924883471092050587230, 5.31637066253986291050518056225, 5.98230698645721555684263097700, 7.14774707141088945815036854833, 8.040211729453307552480174894487, 8.635032155642922128314261978236, 9.292115199073971096801960620782

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