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Properties

Label 2-675-27.22-c1-0-12
Degree $2$
Conductor $675$
Sign $-0.736 - 0.676i$
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  + (0.318 − 0.267i)2-s + (−0.159 + 1.72i)3-s + (−0.317 + 1.79i)4-s + (0.409 + 0.591i)6-s + (0.229 + 1.29i)7-s + (0.795 + 1.37i)8-s + (−2.94 − 0.551i)9-s + (4.90 + 1.78i)11-s + (−3.05 − 0.834i)12-s + (0.0138 + 0.0116i)13-s + (0.419 + 0.352i)14-s + (−2.81 − 1.02i)16-s + (−1.56 + 2.71i)17-s + (−1.08 + 0.612i)18-s + (−0.208 − 0.361i)19-s + ⋯
L(s)  = 1  + (0.225 − 0.188i)2-s + (−0.0922 + 0.995i)3-s + (−0.158 + 0.899i)4-s + (0.167 + 0.241i)6-s + (0.0866 + 0.491i)7-s + (0.281 + 0.486i)8-s + (−0.982 − 0.183i)9-s + (1.47 + 0.537i)11-s + (−0.881 − 0.241i)12-s + (0.00383 + 0.00321i)13-s + (0.112 + 0.0941i)14-s + (−0.703 − 0.256i)16-s + (−0.379 + 0.658i)17-s + (−0.255 + 0.144i)18-s + (−0.0478 − 0.0829i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.736 - 0.676i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.736 - 0.676i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.526185 + 1.35136i\)
\(L(\frac12)\) \(\approx\) \(0.526185 + 1.35136i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.159 - 1.72i)T \)
5 \( 1 \)
good2 \( 1 + (-0.318 + 0.267i)T + (0.347 - 1.96i)T^{2} \)
7 \( 1 + (-0.229 - 1.29i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (-4.90 - 1.78i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (-0.0138 - 0.0116i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (1.56 - 2.71i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.208 + 0.361i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.179 + 1.01i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (5.98 - 5.01i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-0.647 + 3.67i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (-2.21 + 3.83i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.81 + 2.36i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (7.80 + 2.84i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.23 - 6.99i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 - 1.30T + 53T^{2} \)
59 \( 1 + (-3.47 + 1.26i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-1.20 - 6.80i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-8.44 - 7.08i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-3.04 + 5.26i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.273 + 0.473i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.374 + 0.314i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (3.53 - 2.96i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (-1.68 - 2.92i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.34 - 3.40i)T + (74.3 + 62.3i)T^{2} \)
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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

−11.00717936246322104753323230946, −9.887036133675585424427245528359, −8.982490069268780253032083174718, −8.622524780432419390697792374749, −7.35711852094336720378873776941, −6.27957913307692559685186829930, −5.13777903888452829177494697286, −4.14460962669847745671606901376, −3.57491424097319745816096588908, −2.19588921289253512907295279318, 0.75220362470651140699437670473, 1.86222458672926423336553022083, 3.57758720805954885249524263219, 4.79734081651286234769074467305, 5.87205684951414267636878342994, 6.58948285514967127893009252797, 7.22872065819903089238414300040, 8.431020474945202536338984132347, 9.260443139134693426371122439164, 10.14751668783970845519628576095

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