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Properties

Label 2-675-25.11-c1-0-1
Degree $2$
Conductor $675$
Sign $-0.760 + 0.648i$
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  + (−0.324 + 0.235i)2-s + (−0.568 + 1.74i)4-s + (−0.0534 + 2.23i)5-s − 3.13·7-s + (−0.476 − 1.46i)8-s + (−0.509 − 0.738i)10-s + (0.141 − 0.103i)11-s + (−1.01 − 0.738i)13-s + (1.01 − 0.738i)14-s + (−2.47 − 1.79i)16-s + (2.01 + 6.19i)17-s + (−1.51 − 4.67i)19-s + (−3.87 − 1.36i)20-s + (−0.0217 + 0.0669i)22-s + (−2.48 + 1.80i)23-s + ⋯
L(s)  = 1  + (−0.229 + 0.166i)2-s + (−0.284 + 0.874i)4-s + (−0.0239 + 0.999i)5-s − 1.18·7-s + (−0.168 − 0.517i)8-s + (−0.161 − 0.233i)10-s + (0.0427 − 0.0310i)11-s + (−0.281 − 0.204i)13-s + (0.271 − 0.197i)14-s + (−0.618 − 0.449i)16-s + (0.488 + 1.50i)17-s + (−0.348 − 1.07i)19-s + (−0.867 − 0.304i)20-s + (−0.00463 + 0.0142i)22-s + (−0.517 + 0.375i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.760 + 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.760 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.760 + 0.648i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.760 + 0.648i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0908472 - 0.246513i\)
\(L(\frac12)\) \(\approx\) \(0.0908472 - 0.246513i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0534 - 2.23i)T \)
good2 \( 1 + (0.324 - 0.235i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + 3.13T + 7T^{2} \)
11 \( 1 + (-0.141 + 0.103i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.01 + 0.738i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.01 - 6.19i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.51 + 4.67i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (2.48 - 1.80i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-3.10 + 9.54i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.54 + 4.74i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.72 - 1.25i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-2.44 - 1.77i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 6.72T + 43T^{2} \)
47 \( 1 + (3.99 - 12.3i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.37 + 4.23i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (10.5 + 7.68i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (4.64 - 3.37i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (2.19 + 6.75i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (1.96 - 6.04i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (3.56 - 2.59i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.71 - 11.4i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.31 - 7.12i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (6.44 - 4.68i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (4.86 - 14.9i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.98227794780825304216548173734, −9.900879698205694657502035405409, −9.523448393478759843994873371297, −8.230516730709372722450457365746, −7.63738015361296559284958354317, −6.56273504942986539186077453996, −6.08857972589106406487816419534, −4.29599767938791146378051608128, −3.42496626742184322754505450224, −2.56530373104149043263546453186, 0.14842449454794481469159626332, 1.58998017556422243179795670104, 3.18909424656994113450577478359, 4.56573413467339294396568953625, 5.37611108844132727442516438762, 6.26280788777141442643068092393, 7.29886084403266251577707454221, 8.616412602336224674786696780716, 9.168849158721215033477229969105, 9.948269056060291261826730636596

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