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Properties

Label 2-675-135.68-c1-0-26
Degree $2$
Conductor $675$
Sign $-0.105 - 0.994i$
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  + (−1.36 + 1.94i)2-s + (1.63 + 0.583i)3-s + (−1.25 − 3.43i)4-s + (−3.36 + 2.38i)6-s + (2.19 − 1.02i)7-s + (3.81 + 1.02i)8-s + (2.32 + 1.90i)9-s + (0.00275 + 0.00327i)11-s + (−0.0362 − 6.34i)12-s + (1.13 − 0.792i)13-s + (−1.00 + 5.68i)14-s + (−1.59 + 1.33i)16-s + (−1.90 + 0.510i)17-s + (−6.87 + 1.92i)18-s + (6.69 − 3.86i)19-s + ⋯
L(s)  = 1  + (−0.964 + 1.37i)2-s + (0.941 + 0.336i)3-s + (−0.626 − 1.71i)4-s + (−1.37 + 0.972i)6-s + (0.830 − 0.387i)7-s + (1.34 + 0.361i)8-s + (0.773 + 0.633i)9-s + (0.000829 + 0.000988i)11-s + (−0.0104 − 1.83i)12-s + (0.313 − 0.219i)13-s + (−0.267 + 1.51i)14-s + (−0.398 + 0.334i)16-s + (−0.461 + 0.123i)17-s + (−1.61 + 0.453i)18-s + (1.53 − 0.886i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.105 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.105 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.105 - 0.994i$
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: \((2,\ 675,\ (\ :1/2),\ -0.105 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.933235 + 1.03707i\)
\(L(\frac12)\) \(\approx\) \(0.933235 + 1.03707i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.63 - 0.583i)T \)
5 \( 1 \)
good2 \( 1 + (1.36 - 1.94i)T + (-0.684 - 1.87i)T^{2} \)
7 \( 1 + (-2.19 + 1.02i)T + (4.49 - 5.36i)T^{2} \)
11 \( 1 + (-0.00275 - 0.00327i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (-1.13 + 0.792i)T + (4.44 - 12.2i)T^{2} \)
17 \( 1 + (1.90 - 0.510i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-6.69 + 3.86i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.12 + 6.70i)T + (-14.7 - 17.6i)T^{2} \)
29 \( 1 + (-1.74 - 9.87i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (5.62 - 2.04i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-0.451 - 1.68i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.95 + 0.520i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-7.18 - 0.628i)T + (42.3 + 7.46i)T^{2} \)
47 \( 1 + (-0.871 - 1.86i)T + (-30.2 + 36.0i)T^{2} \)
53 \( 1 + (-1.25 + 1.25i)T - 53iT^{2} \)
59 \( 1 + (0.763 + 0.640i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (6.39 + 2.32i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (5.21 + 7.45i)T + (-22.9 + 62.9i)T^{2} \)
71 \( 1 + (-8.32 - 4.80i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.744 - 2.77i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (6.95 - 1.22i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (8.34 + 5.84i)T + (28.3 + 77.9i)T^{2} \)
89 \( 1 + (3.03 + 5.26i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.09 - 12.5i)T + (-95.5 - 16.8i)T^{2} \)
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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

−10.49719678700455692790358965453, −9.374271950994848037807305331388, −8.870557387736146398709945551983, −8.164692539285586520257439460100, −7.35118335842436082736447219191, −6.79302016736963841139341165291, −5.33026865960802097632650017310, −4.60579609247682710338120445752, −3.06590982697925071678348207248, −1.26796561629944835127135526939, 1.25191980665928921888755830875, 2.13445994138232077638285501856, 3.20534470481559922014589902894, 4.14290134201411265888026504491, 5.70605722235965038367395553962, 7.38637757235446998977880549871, 7.927770732453441640476077185560, 8.793600272228715794112866312154, 9.425943208531429301354094200895, 10.05753707972975401070186674401

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