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Properties

Label 2-675-135.113-c1-0-27
Degree $2$
Conductor $675$
Sign $0.691 + 0.722i$
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  + (−0.723 − 0.0632i)2-s + (0.920 + 1.46i)3-s + (−1.45 − 0.255i)4-s + (−0.573 − 1.11i)6-s + (−3.84 − 2.68i)7-s + (2.43 + 0.652i)8-s + (−1.30 + 2.70i)9-s + (−0.808 + 2.22i)11-s + (−0.960 − 2.36i)12-s + (−0.141 − 1.61i)13-s + (2.60 + 2.18i)14-s + (1.04 + 0.381i)16-s + (5.76 − 1.54i)17-s + (1.11 − 1.87i)18-s + (5.87 − 3.39i)19-s + ⋯
L(s)  = 1  + (−0.511 − 0.0447i)2-s + (0.531 + 0.847i)3-s + (−0.725 − 0.127i)4-s + (−0.233 − 0.456i)6-s + (−1.45 − 1.01i)7-s + (0.861 + 0.230i)8-s + (−0.434 + 0.900i)9-s + (−0.243 + 0.670i)11-s + (−0.277 − 0.682i)12-s + (−0.0392 − 0.449i)13-s + (0.696 + 0.584i)14-s + (0.261 + 0.0953i)16-s + (1.39 − 0.374i)17-s + (0.262 − 0.441i)18-s + (1.34 − 0.778i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.691 + 0.722i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (518, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.691 + 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.731421 - 0.312621i\)
\(L(\frac12)\) \(\approx\) \(0.731421 - 0.312621i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.920 - 1.46i)T \)
5 \( 1 \)
good2 \( 1 + (0.723 + 0.0632i)T + (1.96 + 0.347i)T^{2} \)
7 \( 1 + (3.84 + 2.68i)T + (2.39 + 6.57i)T^{2} \)
11 \( 1 + (0.808 - 2.22i)T + (-8.42 - 7.07i)T^{2} \)
13 \( 1 + (0.141 + 1.61i)T + (-12.8 + 2.25i)T^{2} \)
17 \( 1 + (-5.76 + 1.54i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-5.87 + 3.39i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.36 + 6.23i)T + (-7.86 + 21.6i)T^{2} \)
29 \( 1 + (-5.33 + 4.47i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.379 - 2.15i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (1.29 + 4.82i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-3.01 + 3.59i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (-1.37 - 2.94i)T + (-27.6 + 32.9i)T^{2} \)
47 \( 1 + (-5.60 + 8.00i)T + (-16.0 - 44.1i)T^{2} \)
53 \( 1 + (-1.85 + 1.85i)T - 53iT^{2} \)
59 \( 1 + (4.22 - 1.53i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.391 - 2.22i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (7.96 - 0.696i)T + (65.9 - 11.6i)T^{2} \)
71 \( 1 + (0.899 + 0.519i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.991 + 3.70i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (4.12 + 4.91i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (0.0639 - 0.731i)T + (-81.7 - 14.4i)T^{2} \)
89 \( 1 + (0.0118 + 0.0204i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.80 - 1.30i)T + (62.3 - 74.3i)T^{2} \)
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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

−10.15419028650145163659404008039, −9.706886565552955476760885720717, −8.944114277172085236713605853807, −7.84258164119402115006462967913, −7.21933009305998852016461669671, −5.72642761594065216763929109339, −4.67729998503701891365005641334, −3.80182468076304617500367461800, −2.82728524001358239821332287717, −0.55597805776521848587696811958, 1.24900294699790190837083457378, 3.00189059675745257852931879631, 3.62664253272628801657860765679, 5.53247787627983471115624065767, 6.12351103309189083581152860682, 7.39711062773813442112224146325, 8.055022623531592403857470802127, 8.926665328038054389768654697882, 9.570483787078757776003369665579, 10.11929270432396576923949622633

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