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Properties

Label 2-675-45.34-c1-0-12
Degree $2$
Conductor $675$
Sign $-0.891 - 0.452i$
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  + (−2.17 − 1.25i)2-s + (2.16 + 3.74i)4-s + (−0.445 − 0.257i)7-s − 5.83i·8-s + (−1.66 + 2.87i)11-s + (−1.14 + 0.660i)13-s + (0.646 + 1.11i)14-s + (−3.01 + 5.22i)16-s − 3.32i·17-s + 1.32·19-s + (7.23 − 4.17i)22-s + (−3.57 + 2.06i)23-s + 3.32·26-s − 2.22i·28-s + (0.693 − 1.20i)29-s + ⋯
L(s)  = 1  + (−1.53 − 0.888i)2-s + (1.08 + 1.87i)4-s + (−0.168 − 0.0971i)7-s − 2.06i·8-s + (−0.500 + 0.867i)11-s + (−0.317 + 0.183i)13-s + (0.172 + 0.299i)14-s + (−0.753 + 1.30i)16-s − 0.805i·17-s + 0.303·19-s + (1.54 − 0.890i)22-s + (−0.745 + 0.430i)23-s + 0.651·26-s − 0.419i·28-s + (0.128 − 0.222i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.891 - 0.452i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.891 - 0.452i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0217543 + 0.0909555i\)
\(L(\frac12)\) \(\approx\) \(0.0217543 + 0.0909555i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.17 + 1.25i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (0.445 + 0.257i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.66 - 2.87i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.14 - 0.660i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.32iT - 17T^{2} \)
19 \( 1 - 1.32T + 19T^{2} \)
23 \( 1 + (3.57 - 2.06i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.693 + 1.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.36 + 7.56i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.292iT - 37T^{2} \)
41 \( 1 + (5.67 + 9.82i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.96 + 5.17i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.21 - 2.43i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.02iT - 53T^{2} \)
59 \( 1 + (-2.51 - 4.35i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.67 - 6.36i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.18 - 4.72i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.99T + 71T^{2} \)
73 \( 1 - 6.05iT - 73T^{2} \)
79 \( 1 + (4.02 - 6.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.33 - 0.771i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (10.6 + 6.12i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.967083922797498109956960278120, −9.370212184698719464517831756018, −8.494060361816904697175732178956, −7.49187269843401613975086629327, −7.07648730243678711494362509074, −5.47700818811961216121819641534, −4.03901088271195500503943908567, −2.75580272134436470091107095806, −1.79699803932703969552874301354, −0.080618401500632301574706544376, 1.57595661337750880755575752139, 3.20321451115514443644521261771, 5.02567920525743334823766064663, 6.03695048639296715840670703107, 6.69596634151635056897010810341, 7.79538153284797427269614124067, 8.321352307722031333555374982154, 9.087098417152902621813760869359, 10.03814029460545946734715643995, 10.55937300569776684601236649458

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