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Properties

Label 2-675-15.14-c2-0-41
Degree $2$
Conductor $675$
Sign $-0.894 + 0.447i$
Analytic cond. $18.3924$
Root an. cond. $4.28863$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.94·2-s − 0.213·4-s + 6.41i·7-s − 8.19·8-s − 2.74i·11-s − 20.3i·13-s + 12.4i·14-s − 15.1·16-s − 21.0·17-s − 13.6·19-s − 5.34i·22-s + 1.87·23-s − 39.5i·26-s − 1.36i·28-s − 6.98i·29-s + ⋯
L(s)  = 1  + 0.972·2-s − 0.0532·4-s + 0.915i·7-s − 1.02·8-s − 0.249i·11-s − 1.56i·13-s + 0.891i·14-s − 0.943·16-s − 1.24·17-s − 0.719·19-s − 0.243i·22-s + 0.0815·23-s − 1.51i·26-s − 0.0487i·28-s − 0.240i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(18.3924\)
Root analytic conductor: \(4.28863\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (674, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1),\ -0.894 + 0.447i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4597804720\)
\(L(\frac12)\) \(\approx\) \(0.4597804720\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 - 1.94T + 4T^{2} \)
7 \( 1 - 6.41iT - 49T^{2} \)
11 \( 1 + 2.74iT - 121T^{2} \)
13 \( 1 + 20.3iT - 169T^{2} \)
17 \( 1 + 21.0T + 289T^{2} \)
19 \( 1 + 13.6T + 361T^{2} \)
23 \( 1 - 1.87T + 529T^{2} \)
29 \( 1 + 6.98iT - 841T^{2} \)
31 \( 1 + 26.8T + 961T^{2} \)
37 \( 1 - 19.1iT - 1.36e3T^{2} \)
41 \( 1 + 67.2iT - 1.68e3T^{2} \)
43 \( 1 + 67.0iT - 1.84e3T^{2} \)
47 \( 1 + 70.6T + 2.20e3T^{2} \)
53 \( 1 + 81.2T + 2.80e3T^{2} \)
59 \( 1 - 106. iT - 3.48e3T^{2} \)
61 \( 1 - 13.4T + 3.72e3T^{2} \)
67 \( 1 - 63.0iT - 4.48e3T^{2} \)
71 \( 1 - 32.1iT - 5.04e3T^{2} \)
73 \( 1 - 123. iT - 5.32e3T^{2} \)
79 \( 1 - 19.0T + 6.24e3T^{2} \)
83 \( 1 + 119.T + 6.88e3T^{2} \)
89 \( 1 + 152. iT - 7.92e3T^{2} \)
97 \( 1 - 16.4iT - 9.40e3T^{2} \)
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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

−9.981571455010913360503296987162, −8.801219579706209814715620466127, −8.492716675896144306958398999311, −7.06457120718643311972857362540, −5.91441743560920944970735499779, −5.44431818565003905395859705624, −4.39913874639737649256160650857, −3.31017115016063039251938125231, −2.33226660035210045733932523653, −0.11158419284066473873499988288, 1.89054224469789737759337853831, 3.38290950264006769148759611083, 4.41590093797536597467057201315, 4.75453006957967457898573019119, 6.33637137618137013663491157904, 6.73694968579246940729585734026, 8.003211033371720029693406650601, 9.107057801993248042769839289332, 9.655174865294847207740613555370, 11.00353914106403170755110920647

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