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Properties

Label 2-675-45.29-c2-0-24
Degree $2$
Conductor $675$
Sign $0.397 + 0.917i$
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  + (−0.866 + 1.5i)2-s + (0.5 + 0.866i)4-s + (−1.73 − i)7-s − 8.66·8-s + (1.5 + 0.866i)11-s + (3.46 − 2i)13-s + (3 − 1.73i)14-s + (5.5 − 9.52i)16-s − 15.5·17-s − 11·19-s + (−2.59 + 1.5i)22-s + (−13.8 − 24i)23-s + 6.92i·26-s − 1.99i·28-s + (39 + 22.5i)29-s + ⋯
L(s)  = 1  + (−0.433 + 0.750i)2-s + (0.125 + 0.216i)4-s + (−0.247 − 0.142i)7-s − 1.08·8-s + (0.136 + 0.0787i)11-s + (0.266 − 0.153i)13-s + (0.214 − 0.123i)14-s + (0.343 − 0.595i)16-s − 0.916·17-s − 0.578·19-s + (−0.118 + 0.0681i)22-s + (−0.602 − 1.04i)23-s + 0.266i·26-s − 0.0714i·28-s + (1.34 + 0.776i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.397 + 0.917i$
Analytic conductor: \(18.3924\)
Root analytic conductor: \(4.28863\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1),\ 0.397 + 0.917i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4680461143\)
\(L(\frac12)\) \(\approx\) \(0.4680461143\)
\(L(2)\) 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 \( 1 + (0.866 - 1.5i)T + (-2 - 3.46i)T^{2} \)
7 \( 1 + (1.73 + i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-1.5 - 0.866i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-3.46 + 2i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 15.5T + 289T^{2} \)
19 \( 1 + 11T + 361T^{2} \)
23 \( 1 + (13.8 + 24i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-39 - 22.5i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (16 + 27.7i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 34iT - 1.36e3T^{2} \)
41 \( 1 + (-10.5 + 6.06i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (52.8 + 30.5i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (24.2 - 42i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 2.80e3T^{2} \)
59 \( 1 + (-43.5 + 25.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (28 - 48.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (26.8 - 15.5i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 31.1iT - 5.04e3T^{2} \)
73 \( 1 + 65iT - 5.32e3T^{2} \)
79 \( 1 + (-19 + 32.9i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-24.2 + 42i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 124. iT - 7.92e3T^{2} \)
97 \( 1 + (-99.5 - 57.5i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.03280036651972129904540850171, −8.926978635033063350840277108934, −8.462810519438717637984853546958, −7.47448763840095768248076018367, −6.62028562316677200611949886421, −6.04422294713509368932549003933, −4.64616607813556828975318751787, −3.49346284978654208141618988974, −2.25099055376131793172322944289, −0.19054130960114348180274182500, 1.38215162403296979243710149630, 2.50550586635882628331619592863, 3.62307272235005626315194993926, 4.93530277628388974422729670717, 6.16549379429528960805616512909, 6.71424522413889093613628261432, 8.153159205134682356484982470880, 8.913028735225571282298071410996, 9.757152018747200551966416692533, 10.39030560425841080765528477522

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