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Properties

Label 2-675-45.4-c1-0-12
Degree $2$
Conductor $675$
Sign $0.349 + 0.936i$
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  + (1.80 − 1.04i)2-s + (1.17 − 2.03i)4-s + (3.53 − 2.04i)7-s − 0.734i·8-s + (−0.675 − 1.17i)11-s + (0.561 + 0.324i)13-s + (4.26 − 7.38i)14-s + (1.58 + 2.74i)16-s + 1.35i·17-s − 0.648·19-s + (−2.44 − 1.41i)22-s + (−4.14 − 2.39i)23-s + 1.35·26-s − 9.61i·28-s + (−1.93 − 3.35i)29-s + ⋯
L(s)  = 1  + (1.27 − 0.737i)2-s + (0.587 − 1.01i)4-s + (1.33 − 0.772i)7-s − 0.259i·8-s + (−0.203 − 0.353i)11-s + (0.155 + 0.0898i)13-s + (1.13 − 1.97i)14-s + (0.396 + 0.686i)16-s + 0.327i·17-s − 0.148·19-s + (−0.520 − 0.300i)22-s + (−0.864 − 0.499i)23-s + 0.265·26-s − 1.81i·28-s + (−0.359 − 0.623i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.349 + 0.936i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.349 + 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.66375 - 1.84916i\)
\(L(\frac12)\) \(\approx\) \(2.66375 - 1.84916i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2 \( 1 + (-1.80 + 1.04i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-3.53 + 2.04i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.675 + 1.17i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.561 - 0.324i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.35iT - 17T^{2} \)
19 \( 1 + 0.648T + 19T^{2} \)
23 \( 1 + (4.14 + 2.39i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.93 + 3.35i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.84 + 6.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.52iT - 37T^{2} \)
41 \( 1 + (0.0898 - 0.155i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.710 + 0.410i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (9.44 - 5.45i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 4.17iT - 53T^{2} \)
59 \( 1 + (2.08 - 3.61i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.91 - 3.30i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.05 + 4.07i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.11T + 71T^{2} \)
73 \( 1 - 12.3iT - 73T^{2} \)
79 \( 1 + (-5.17 - 8.95i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.6 - 6.12i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (11.7 - 6.79i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.72364439712270638211942298636, −9.868376813780100243994603858888, −8.321383144611295800370336052410, −7.889763056840845107618626948935, −6.45209743747772355337647248423, −5.50494711451438173156029627645, −4.49997241040816136379007701497, −4.00076726723336806698638423085, −2.63918021333472390230592662525, −1.45380840350762062427866848379, 1.90537150648926827379441457099, 3.33133273454539873376821740886, 4.55109843741998751251727364777, 5.17288408961574865803990569822, 5.91999909895473999486953134907, 6.99876040425755911473278537828, 7.84689429929031203225207070686, 8.649483040111415978855599694593, 9.791924305962516009009206164211, 10.92008922099546550026015486344

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