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Properties

Label 2-675-135.68-c1-0-48
Degree $2$
Conductor $675$
Sign $-0.263 - 0.964i$
Analytic cond. $5.38990$
Root an. cond. $2.32161$
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  + (1.52 − 2.18i)2-s + (−1.25 + 1.19i)3-s + (−1.73 − 4.77i)4-s + (0.674 + 4.56i)6-s + (−2.78 + 1.30i)7-s + (−7.92 − 2.12i)8-s + (0.166 − 2.99i)9-s + (−0.426 − 0.508i)11-s + (7.87 + 3.94i)12-s + (−0.384 + 0.269i)13-s + (−1.42 + 8.06i)14-s + (−8.94 + 7.50i)16-s + (−4.50 + 1.20i)17-s + (−6.27 − 4.93i)18-s + (−4.80 + 2.77i)19-s + ⋯
L(s)  = 1  + (1.07 − 1.54i)2-s + (−0.726 + 0.687i)3-s + (−0.869 − 2.38i)4-s + (0.275 + 1.86i)6-s + (−1.05 + 0.491i)7-s + (−2.80 − 0.750i)8-s + (0.0555 − 0.998i)9-s + (−0.128 − 0.153i)11-s + (2.27 + 1.13i)12-s + (−0.106 + 0.0746i)13-s + (−0.380 + 2.15i)14-s + (−2.23 + 1.87i)16-s + (−1.09 + 0.292i)17-s + (−1.47 − 1.16i)18-s + (−1.10 + 0.637i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.263 - 0.964i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.263 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.207265 + 0.271464i\)
\(L(\frac12)\) \(\approx\) \(0.207265 + 0.271464i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.25 - 1.19i)T \)
5 \( 1 \)
good2 \( 1 + (-1.52 + 2.18i)T + (-0.684 - 1.87i)T^{2} \)
7 \( 1 + (2.78 - 1.30i)T + (4.49 - 5.36i)T^{2} \)
11 \( 1 + (0.426 + 0.508i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (0.384 - 0.269i)T + (4.44 - 12.2i)T^{2} \)
17 \( 1 + (4.50 - 1.20i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (4.80 - 2.77i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.0280 - 0.0602i)T + (-14.7 - 17.6i)T^{2} \)
29 \( 1 + (0.434 + 2.46i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-1.76 + 0.642i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (0.656 + 2.44i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.62 - 0.286i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (8.36 + 0.732i)T + (42.3 + 7.46i)T^{2} \)
47 \( 1 + (1.73 + 3.71i)T + (-30.2 + 36.0i)T^{2} \)
53 \( 1 + (-7.52 + 7.52i)T - 53iT^{2} \)
59 \( 1 + (-3.49 - 2.93i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (5.84 + 2.12i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (3.13 + 4.48i)T + (-22.9 + 62.9i)T^{2} \)
71 \( 1 + (5.33 + 3.07i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.64 + 13.6i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.34 + 0.941i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (4.64 + 3.25i)T + (28.3 + 77.9i)T^{2} \)
89 \( 1 + (-3.08 - 5.33i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.18 + 13.5i)T + (-95.5 - 16.8i)T^{2} \)
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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

−10.25820606948517053495639795181, −9.513998535575079742645767616427, −8.732995934815772925269552290514, −6.50484594375668924023447464322, −6.01223092902696104735208000650, −4.99627574744966555039219488117, −4.10899334877248002567558735803, −3.31419297190079457702702110870, −2.11506103036492757840675040014, −0.12973839735175194444695119112, 2.75111923196652864985116834060, 4.14333389086590823752835598355, 4.93960071700940304363061291416, 5.95984222832275434037803502055, 6.77715491183729487915097768258, 6.98362428334074148401666307012, 8.091725585462489522595015818228, 8.986178813693678144048801306877, 10.31856595575051358820127459209, 11.41268752495389214340443624785

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