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Properties

Label 2-1740-145.128-c1-0-23
Degree $2$
Conductor $1740$
Sign $-0.364 + 0.931i$
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 + (0.0712 − 2.23i)5-s + (−0.799 + 0.799i)7-s − 9-s + (−0.319 − 0.319i)11-s + (1.26 − 1.26i)13-s + (2.23 + 0.0712i)15-s − 3.94·17-s + (2.75 − 2.75i)19-s + (−0.799 − 0.799i)21-s + (0.270 + 0.270i)23-s + (−4.98 − 0.318i)25-s − i·27-s + (0.120 + 5.38i)29-s + (−3.45 − 3.45i)31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.0318 − 0.999i)5-s + (−0.302 + 0.302i)7-s − 0.333·9-s + (−0.0963 − 0.0963i)11-s + (0.351 − 0.351i)13-s + (0.577 + 0.0183i)15-s − 0.957·17-s + (0.633 − 0.633i)19-s + (−0.174 − 0.174i)21-s + (0.0564 + 0.0564i)23-s + (−0.997 − 0.0636i)25-s − 0.192i·27-s + (0.0224 + 0.999i)29-s + (−0.619 − 0.619i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.364 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1740\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 29\)
Sign: $-0.364 + 0.931i$
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: \((2,\ 1740,\ (\ :1/2),\ -0.364 + 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8877699615\)
\(L(\frac12)\) \(\approx\) \(0.8877699615\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0712 + 2.23i)T \)
29 \( 1 + (-0.120 - 5.38i)T \)
good7 \( 1 + (0.799 - 0.799i)T - 7iT^{2} \)
11 \( 1 + (0.319 + 0.319i)T + 11iT^{2} \)
13 \( 1 + (-1.26 + 1.26i)T - 13iT^{2} \)
17 \( 1 + 3.94T + 17T^{2} \)
19 \( 1 + (-2.75 + 2.75i)T - 19iT^{2} \)
23 \( 1 + (-0.270 - 0.270i)T + 23iT^{2} \)
31 \( 1 + (3.45 + 3.45i)T + 31iT^{2} \)
37 \( 1 + 9.42iT - 37T^{2} \)
41 \( 1 + (5.22 - 5.22i)T - 41iT^{2} \)
43 \( 1 + 9.82iT - 43T^{2} \)
47 \( 1 + 5.91iT - 47T^{2} \)
53 \( 1 + (3.04 + 3.04i)T + 53iT^{2} \)
59 \( 1 + 3.40iT - 59T^{2} \)
61 \( 1 + (8.02 + 8.02i)T + 61iT^{2} \)
67 \( 1 + (2.10 + 2.10i)T + 67iT^{2} \)
71 \( 1 + 4.89iT - 71T^{2} \)
73 \( 1 - 1.74T + 73T^{2} \)
79 \( 1 + (-2.41 + 2.41i)T - 79iT^{2} \)
83 \( 1 + (1.43 + 1.43i)T + 83iT^{2} \)
89 \( 1 + (6.19 - 6.19i)T - 89iT^{2} \)
97 \( 1 - 3.50iT - 97T^{2} \)
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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

−9.115991113616938839976319952406, −8.534411292264736344403770206884, −7.61282653361613586103115826022, −6.58063878153319838019316762074, −5.57864202602923411744539687716, −5.04433804898810125236896485610, −4.09835862864191388039400310937, −3.18203142485058636023645884265, −1.90319027909331327517005101183, −0.32493831267994710394086151666, 1.50586986630695009211615475997, 2.64630618672715098088495296078, 3.49919597537884952555126396981, 4.53109160023900107415574967486, 5.80606379001425816007509700924, 6.47967852563195767532247330158, 7.09610255655956490910435738548, 7.82875966911018073742279938118, 8.682447720203994219219108840650, 9.662912059021354313164931299294

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