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Properties

Label 2-675-9.5-c2-0-15
Degree $2$
Conductor $675$
Sign $0.930 - 0.367i$
Analytic cond. $18.3924$
Root an. cond. $4.28863$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.46 − 1.42i)2-s + (2.03 − 3.52i)4-s + (4.86 + 8.42i)7-s − 0.212i·8-s + (−0.370 + 0.214i)11-s + (−9.37 + 16.2i)13-s + (23.9 + 13.8i)14-s + (7.84 + 13.5i)16-s − 2.85i·17-s − 0.530·19-s + (−0.608 + 1.05i)22-s + (18.8 + 10.8i)23-s + 53.2i·26-s + 39.6·28-s + (−21.0 + 12.1i)29-s + ⋯
L(s)  = 1  + (1.23 − 0.710i)2-s + (0.509 − 0.882i)4-s + (0.694 + 1.20i)7-s − 0.0265i·8-s + (−0.0337 + 0.0194i)11-s + (−0.721 + 1.24i)13-s + (1.71 + 0.987i)14-s + (0.490 + 0.849i)16-s − 0.168i·17-s − 0.0279·19-s + (−0.0276 + 0.0479i)22-s + (0.819 + 0.473i)23-s + 2.04i·26-s + 1.41·28-s + (−0.726 + 0.419i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 - 0.367i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.930 - 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.930 - 0.367i$
Analytic conductor: \(18.3924\)
Root analytic conductor: \(4.28863\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (476, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1),\ 0.930 - 0.367i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.543023422\)
\(L(\frac12)\) \(\approx\) \(3.543023422\)
\(L(2)\) 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 \( 1 \)
good2 \( 1 + (-2.46 + 1.42i)T + (2 - 3.46i)T^{2} \)
7 \( 1 + (-4.86 - 8.42i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (0.370 - 0.214i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (9.37 - 16.2i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 2.85iT - 289T^{2} \)
19 \( 1 + 0.530T + 361T^{2} \)
23 \( 1 + (-18.8 - 10.8i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (21.0 - 12.1i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-6.33 + 10.9i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 14.5T + 1.36e3T^{2} \)
41 \( 1 + (-33.1 - 19.1i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (28.8 + 50.0i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-42.9 + 24.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 44.5iT - 2.80e3T^{2} \)
59 \( 1 + (54.6 + 31.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-11.0 - 19.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (16.2 - 28.1i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 89.8iT - 5.04e3T^{2} \)
73 \( 1 - 144.T + 5.32e3T^{2} \)
79 \( 1 + (-25.1 - 43.5i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-66.2 + 38.2i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 28.9iT - 7.92e3T^{2} \)
97 \( 1 + (11.4 + 19.9i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.77293004252287224735323848081, −9.475014833389862986149287820643, −8.796475221567554790016032417787, −7.69617253860998571666861073194, −6.49454825289526490891718469469, −5.38013725980059066675309752020, −4.88543225075190769304703438819, −3.81187959152343173260970942664, −2.56420600372668114287664502154, −1.82356801042679196007936017925, 0.876028799275523885567374825456, 2.85847699476808379165818441325, 4.00619199940515678289413178854, 4.76731557047497893321527793918, 5.55767266360359220244319216329, 6.60767970331519272609886300201, 7.52069226591017446753892665266, 7.962703006939213536485985363504, 9.447306897326959798348190608040, 10.44122775453212728040694754175

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