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Properties

Label 2-675-135.83-c1-0-31
Degree $2$
Conductor $675$
Sign $0.600 - 0.799i$
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  + (−1.65 + 1.15i)2-s + (1.69 + 0.352i)3-s + (0.704 − 1.93i)4-s + (−3.20 + 1.37i)6-s + (−0.618 + 1.32i)7-s + (0.0310 + 0.115i)8-s + (2.75 + 1.19i)9-s + (1.86 − 2.21i)11-s + (1.87 − 3.03i)12-s + (3.51 − 5.02i)13-s + (−0.512 − 2.90i)14-s + (2.97 + 2.49i)16-s + (0.833 − 3.10i)17-s + (−5.92 + 1.20i)18-s + (3.51 + 2.03i)19-s + ⋯
L(s)  = 1  + (−1.16 + 0.817i)2-s + (0.979 + 0.203i)3-s + (0.352 − 0.967i)4-s + (−1.30 + 0.562i)6-s + (−0.233 + 0.501i)7-s + (0.0109 + 0.0409i)8-s + (0.916 + 0.398i)9-s + (0.560 − 0.668i)11-s + (0.541 − 0.875i)12-s + (0.975 − 1.39i)13-s + (−0.136 − 0.776i)14-s + (0.742 + 0.623i)16-s + (0.202 − 0.753i)17-s + (−1.39 + 0.283i)18-s + (0.806 + 0.465i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.600 - 0.799i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.600 - 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09974 + 0.549688i\)
\(L(\frac12)\) \(\approx\) \(1.09974 + 0.549688i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.69 - 0.352i)T \)
5 \( 1 \)
good2 \( 1 + (1.65 - 1.15i)T + (0.684 - 1.87i)T^{2} \)
7 \( 1 + (0.618 - 1.32i)T + (-4.49 - 5.36i)T^{2} \)
11 \( 1 + (-1.86 + 2.21i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-3.51 + 5.02i)T + (-4.44 - 12.2i)T^{2} \)
17 \( 1 + (-0.833 + 3.10i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-3.51 - 2.03i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.58 - 2.14i)T + (14.7 - 17.6i)T^{2} \)
29 \( 1 + (-0.368 + 2.08i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (3.20 + 1.16i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (-11.3 - 3.04i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.839 + 0.147i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (-0.0641 - 0.732i)T + (-42.3 + 7.46i)T^{2} \)
47 \( 1 + (2.61 + 1.21i)T + (30.2 + 36.0i)T^{2} \)
53 \( 1 + (-0.0757 + 0.0757i)T - 53iT^{2} \)
59 \( 1 + (2.89 - 2.43i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (4.21 - 1.53i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (3.65 + 2.55i)T + (22.9 + 62.9i)T^{2} \)
71 \( 1 + (7.84 - 4.53i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-8.04 + 2.15i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-1.44 - 0.254i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (-9.69 - 13.8i)T + (-28.3 + 77.9i)T^{2} \)
89 \( 1 + (-3.05 + 5.28i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (16.1 - 1.41i)T + (95.5 - 16.8i)T^{2} \)
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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

−10.11483924178346151776055456999, −9.498955673322373019136308320413, −8.854315807507544174552230207159, −7.974318078543613170436433899172, −7.66062757935539386668870729913, −6.32864694176886767209404971700, −5.60059907558349084192107650735, −3.84605980932595045376710395271, −2.94761122135210467563878313489, −1.11640629560463930985913687780, 1.26258933412680501848703134912, 2.13796694013287563097755589086, 3.48426379716393214422409995487, 4.34688829758673743041377003961, 6.25504638912845753484454614102, 7.22412270010880816553627394557, 8.029086105767596532521739209445, 8.950116333583398496480513659685, 9.401524173307376766552694516886, 10.13823499498330410289048644277

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