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Properties

Label 2-675-225.31-c1-0-15
Degree $2$
Conductor $675$
Sign $0.897 + 0.441i$
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  + (−0.721 + 0.800i)2-s + (0.0876 + 0.834i)4-s + (−2.23 − 0.0235i)5-s + (0.316 + 0.548i)7-s + (−2.47 − 1.79i)8-s + (1.63 − 1.77i)10-s + (3.49 − 3.87i)11-s + (−2.54 − 2.82i)13-s + (−0.667 − 0.141i)14-s + (1.58 − 0.336i)16-s + (0.365 + 0.265i)17-s + (−3.97 − 2.88i)19-s + (−0.176 − 1.86i)20-s + (0.587 + 5.59i)22-s + (−0.100 − 0.0213i)23-s + ⋯
L(s)  = 1  + (−0.509 + 0.566i)2-s + (0.0438 + 0.417i)4-s + (−0.999 − 0.0105i)5-s + (0.119 + 0.207i)7-s + (−0.874 − 0.635i)8-s + (0.515 − 0.560i)10-s + (1.05 − 1.16i)11-s + (−0.706 − 0.784i)13-s + (−0.178 − 0.0378i)14-s + (0.395 − 0.0841i)16-s + (0.0886 + 0.0643i)17-s + (−0.911 − 0.662i)19-s + (−0.0394 − 0.417i)20-s + (0.125 + 1.19i)22-s + (−0.0209 − 0.00444i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 + 0.441i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.897 + 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.897 + 0.441i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.897 + 0.441i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.702165 - 0.163545i\)
\(L(\frac12)\) \(\approx\) \(0.702165 - 0.163545i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (2.23 + 0.0235i)T \)
good2 \( 1 + (0.721 - 0.800i)T + (-0.209 - 1.98i)T^{2} \)
7 \( 1 + (-0.316 - 0.548i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.49 + 3.87i)T + (-1.14 - 10.9i)T^{2} \)
13 \( 1 + (2.54 + 2.82i)T + (-1.35 + 12.9i)T^{2} \)
17 \( 1 + (-0.365 - 0.265i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (3.97 + 2.88i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (0.100 + 0.0213i)T + (21.0 + 9.35i)T^{2} \)
29 \( 1 + (-9.17 - 4.08i)T + (19.4 + 21.5i)T^{2} \)
31 \( 1 + (-3.69 + 1.64i)T + (20.7 - 23.0i)T^{2} \)
37 \( 1 + (-0.0800 + 0.246i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (5.23 + 5.81i)T + (-4.28 + 40.7i)T^{2} \)
43 \( 1 + (-3.68 - 6.38i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (9.67 + 4.30i)T + (31.4 + 34.9i)T^{2} \)
53 \( 1 + (-4.56 + 3.31i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (4.79 + 5.32i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (-3.08 + 3.42i)T + (-6.37 - 60.6i)T^{2} \)
67 \( 1 + (-3.18 + 1.41i)T + (44.8 - 49.7i)T^{2} \)
71 \( 1 + (-3.47 + 2.52i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.75 + 14.6i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1.24 - 0.555i)T + (52.8 + 58.7i)T^{2} \)
83 \( 1 + (-0.223 + 2.12i)T + (-81.1 - 17.2i)T^{2} \)
89 \( 1 + (-5.42 - 16.7i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (5.56 + 2.47i)T + (64.9 + 72.0i)T^{2} \)
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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

−10.42379274621018579915049441790, −9.221842093480511377894448544681, −8.422740825749629963909964556050, −8.094430606794425095253642391425, −6.92823002392102768859854911502, −6.36663957007666358032190609666, −4.93386837691314161325825515879, −3.75007600645177019814606676375, −2.90537199177287800266464976195, −0.50729522334427219339545720476, 1.29572269920280986736005213728, 2.56928050852306790015620094682, 4.12363953167506085987157990087, 4.75908605920581703854871381151, 6.33904829027766953149807117189, 7.03172146519703488395630302713, 8.148575842177647759416915979569, 8.950153856202663930499201747158, 9.871723803050031880743421340153, 10.40741912945018197185413357199

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