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Properties

Label 2-675-27.7-c1-0-48
Degree $2$
Conductor $675$
Sign $-0.835 - 0.549i$
Analytic cond. $5.38990$
Root an. cond. $2.32161$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.826 + 0.300i)2-s + (0.592 − 1.62i)3-s + (−0.939 + 0.788i)4-s + 1.52i·6-s + (−2.87 − 2.41i)7-s + (1.41 − 2.45i)8-s + (−2.29 − 1.92i)9-s + (−0.180 + 1.02i)11-s + (0.726 + 1.99i)12-s + (2.99 + 1.08i)13-s + (3.10 + 1.13i)14-s + (−0.00727 + 0.0412i)16-s + (−0.233 − 0.405i)17-s + (2.47 + 0.902i)18-s + (−2.34 + 4.06i)19-s + ⋯
L(s)  = 1  + (−0.584 + 0.212i)2-s + (0.342 − 0.939i)3-s + (−0.469 + 0.394i)4-s + 0.621i·6-s + (−1.08 − 0.913i)7-s + (0.501 − 0.868i)8-s + (−0.766 − 0.642i)9-s + (−0.0545 + 0.309i)11-s + (0.209 + 0.576i)12-s + (0.830 + 0.302i)13-s + (0.830 + 0.302i)14-s + (−0.00181 + 0.0103i)16-s + (−0.0567 − 0.0982i)17-s + (0.584 + 0.212i)18-s + (−0.538 + 0.932i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.835 - 0.549i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.835 - 0.549i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.592 + 1.62i)T \)
5 \( 1 \)
good2 \( 1 + (0.826 - 0.300i)T + (1.53 - 1.28i)T^{2} \)
7 \( 1 + (2.87 + 2.41i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (0.180 - 1.02i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (-2.99 - 1.08i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (0.233 + 0.405i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.34 - 4.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.11 - 3.45i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (5.45 - 1.98i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (3.14 - 2.63i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (-2.23 - 3.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.52 + 2.73i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-2.11 + 11.9i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (2.65 + 2.22i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 8.83T + 53T^{2} \)
59 \( 1 + (-2.36 - 13.4i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-7.46 - 6.26i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (1.71 + 0.623i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (3.85 + 6.67i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.407 + 0.705i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.81 + 1.38i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (15.9 - 5.81i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-5.19 + 9.00i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.06 + 6.02i)T + (-91.1 - 33.1i)T^{2} \)
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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

−9.815242838639603366258973141456, −8.979493290676816420947654172167, −8.244974253932667295273491889207, −7.32298780234918389023675800660, −6.83246891799800147289380519979, −5.78946081438974505556396900205, −3.97683593028085985336245776085, −3.41549439983230138481795264119, −1.57965517537069703265762732360, 0, 2.26685790394321552586355868351, 3.41501094099480271649277250043, 4.53955848223434342618902710574, 5.63749118667051617947863283401, 6.30494894614235658450182959236, 8.086805866626885886496061631208, 8.630962218181661175103712127717, 9.508230002400118159576195474376, 9.771254242509671863606291340351

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