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Properties

Label 2-675-135.122-c1-0-32
Degree $2$
Conductor $675$
Sign $0.975 - 0.219i$
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  + (1.22 + 0.856i)2-s + (1.44 − 0.960i)3-s + (0.0790 + 0.217i)4-s + (2.58 + 0.0591i)6-s + (2.03 + 4.36i)7-s + (0.683 − 2.55i)8-s + (1.15 − 2.76i)9-s + (−2.25 − 2.68i)11-s + (0.322 + 0.237i)12-s + (2.18 + 3.11i)13-s + (−1.24 + 7.08i)14-s + (3.37 − 2.83i)16-s + (0.367 + 1.37i)17-s + (3.78 − 2.39i)18-s + (1.30 − 0.750i)19-s + ⋯
L(s)  = 1  + (0.865 + 0.605i)2-s + (0.832 − 0.554i)3-s + (0.0395 + 0.108i)4-s + (1.05 + 0.0241i)6-s + (0.769 + 1.65i)7-s + (0.241 − 0.902i)8-s + (0.384 − 0.923i)9-s + (−0.678 − 0.808i)11-s + (0.0931 + 0.0684i)12-s + (0.605 + 0.864i)13-s + (−0.333 + 1.89i)14-s + (0.844 − 0.708i)16-s + (0.0891 + 0.332i)17-s + (0.892 − 0.565i)18-s + (0.298 − 0.172i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.219i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.975 - 0.219i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.975 - 0.219i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.16880 + 0.352159i\)
\(L(\frac12)\) \(\approx\) \(3.16880 + 0.352159i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.44 + 0.960i)T \)
5 \( 1 \)
good2 \( 1 + (-1.22 - 0.856i)T + (0.684 + 1.87i)T^{2} \)
7 \( 1 + (-2.03 - 4.36i)T + (-4.49 + 5.36i)T^{2} \)
11 \( 1 + (2.25 + 2.68i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (-2.18 - 3.11i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (-0.367 - 1.37i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.30 + 0.750i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.35 + 0.633i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (0.168 + 0.957i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (2.44 - 0.891i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (6.69 - 1.79i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.670 + 0.118i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-0.0175 + 0.200i)T + (-42.3 - 7.46i)T^{2} \)
47 \( 1 + (-7.89 + 3.68i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (2.81 + 2.81i)T + 53iT^{2} \)
59 \( 1 + (-5.69 - 4.77i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.08 + 0.396i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (12.5 - 8.79i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (11.1 + 6.42i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.343 - 0.0920i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-9.66 + 1.70i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (6.51 - 9.30i)T + (-28.3 - 77.9i)T^{2} \)
89 \( 1 + (-2.09 - 3.63i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.41 + 0.561i)T + (95.5 + 16.8i)T^{2} \)
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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

−10.55845676747880420823185534623, −9.276961183705419951420038666291, −8.688640371251878252383499228763, −7.939361729387025338869585379002, −6.83456956636982708065994675570, −5.91857042151755488890357104778, −5.30468478852317694152479912527, −4.06048329611704835256459331126, −2.87955236172659574624359821978, −1.65178863885721150469000638484, 1.71005544086694753629674076298, 3.04532557360646656607621355819, 3.90188883719251967912590412394, 4.60436966329198113559447138558, 5.40866626918269545117299771330, 7.38151611866637293031959823827, 7.73891195881942961713975453754, 8.646620841529817465609489721000, 9.953567143001160975952094769870, 10.61625444212108601996462055350

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