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Properties

Label 2-3564-9.4-c1-0-34
Degree $2$
Conductor $3564$
Sign $-0.342 + 0.939i$
Analytic cond. $28.4586$
Root an. cond. $5.33466$
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  + (2.03 − 3.51i)5-s + (0.0326 + 0.0564i)7-s + (0.5 + 0.866i)11-s + (−0.665 + 1.15i)13-s + 2.03·17-s − 0.0245·19-s + (1.08 − 1.88i)23-s + (−5.75 − 9.96i)25-s + (−5.00 − 8.67i)29-s + (−2.22 + 3.84i)31-s + 0.264·35-s + 1.48·37-s + (3.77 − 6.54i)41-s + (−4.82 − 8.36i)43-s + (6.16 + 10.6i)47-s + ⋯
L(s)  = 1  + (0.908 − 1.57i)5-s + (0.0123 + 0.0213i)7-s + (0.150 + 0.261i)11-s + (−0.184 + 0.319i)13-s + 0.494·17-s − 0.00562·19-s + (0.226 − 0.393i)23-s + (−1.15 − 1.99i)25-s + (−0.929 − 1.61i)29-s + (−0.399 + 0.691i)31-s + 0.0447·35-s + 0.243·37-s + (0.589 − 1.02i)41-s + (−0.736 − 1.27i)43-s + (0.899 + 1.55i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3564\)    =    \(2^{2} \cdot 3^{4} \cdot 11\)
Sign: $-0.342 + 0.939i$
Analytic conductor: \(28.4586\)
Root analytic conductor: \(5.33466\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3564} (2377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3564,\ (\ :1/2),\ -0.342 + 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.988729249\)
\(L(\frac12)\) \(\approx\) \(1.988729249\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-2.03 + 3.51i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.0326 - 0.0564i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (0.665 - 1.15i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2.03T + 17T^{2} \)
19 \( 1 + 0.0245T + 19T^{2} \)
23 \( 1 + (-1.08 + 1.88i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.00 + 8.67i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.22 - 3.84i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.48T + 37T^{2} \)
41 \( 1 + (-3.77 + 6.54i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.82 + 8.36i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.16 - 10.6i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.61T + 53T^{2} \)
59 \( 1 + (-5.03 + 8.72i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.07 - 7.05i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.68 + 2.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.101T + 71T^{2} \)
73 \( 1 + 8.12T + 73T^{2} \)
79 \( 1 + (3.66 + 6.34i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.94 - 6.84i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.48T + 89T^{2} \)
97 \( 1 + (4.31 + 7.46i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.561260139004967367003718198981, −7.68238115445616030204654277897, −6.84057628030516283457924099388, −5.77660130614338938589385744518, −5.46000293174204520390961338402, −4.53943115322292578240919084246, −3.89904682553462478257240849318, −2.42395976916956200904739457018, −1.66332045257887161063350328928, −0.58566770226343646225874104156, 1.39778988897770303492251805971, 2.48072097736529729298034757004, 3.14056874666315012268805920816, 3.93911488782575738225256344601, 5.32746969458714077170175276733, 5.78068549418664330754745679198, 6.62394445725701577713967256832, 7.20166013893162620598505834374, 7.84808769363123578641124073893, 8.930949343814712761968428512914

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