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Properties

Label 2-675-45.29-c2-0-15
Degree $2$
Conductor $675$
Sign $0.972 + 0.232i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.232i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.972 + 0.232i$
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: \((2,\ 675,\ (\ :1),\ 0.972 + 0.232i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.782322991\)
\(L(\frac12)\) \(\approx\) \(2.782322991\)
\(L(2)\) 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

−10.44333721709072361340903916697, −9.618694331497994845275745017881, −8.503154656533269877413245230690, −7.64277055169506693314256036618, −6.81456842453914783676339729684, −5.51110819097294956399700896091, −4.54414654884984440881469789841, −3.54415940997415242621371789991, −2.54543248577113764944421340140, −1.32682411377770935763614739414, 1.03236519730993921824488666025, 2.54373239596287692962790525008, 4.09394992405294935900940930472, 4.96754203831998039229330709319, 5.87309094013018994915871040363, 6.70063977198981911396436559793, 7.54260328593107970486434470102, 8.355194943351841211002616713540, 9.465989404014641471186849640573, 10.51400710070898036895651045298

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