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Properties

Label 2-675-9.7-c1-0-11
Degree $2$
Conductor $675$
Sign $0.906 + 0.422i$
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  + (0.258 − 0.448i)2-s + (0.866 + 1.5i)4-s + (1.67 − 2.89i)7-s + 1.93·8-s + (0.633 − 1.09i)11-s + (−1.22 − 2.12i)13-s + (−0.866 − 1.5i)14-s + (−1.23 + 2.13i)16-s + 5.27·17-s − 0.732·19-s + (−0.328 − 0.568i)22-s + (−0.258 − 0.448i)23-s − 1.26·26-s + 5.79·28-s + (−0.232 + 0.401i)29-s + ⋯
L(s)  = 1  + (0.183 − 0.316i)2-s + (0.433 + 0.750i)4-s + (0.632 − 1.09i)7-s + 0.683·8-s + (0.191 − 0.331i)11-s + (−0.339 − 0.588i)13-s + (−0.231 − 0.400i)14-s + (−0.308 + 0.533i)16-s + 1.28·17-s − 0.167·19-s + (−0.0699 − 0.121i)22-s + (−0.0539 − 0.0934i)23-s − 0.248·26-s + 1.09·28-s + (−0.0430 + 0.0746i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.906 + 0.422i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.906 + 0.422i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99478 - 0.442233i\)
\(L(\frac12)\) \(\approx\) \(1.99478 - 0.442233i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.258 + 0.448i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (-1.67 + 2.89i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.633 + 1.09i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.22 + 2.12i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.27T + 17T^{2} \)
19 \( 1 + 0.732T + 19T^{2} \)
23 \( 1 + (0.258 + 0.448i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.232 - 0.401i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.366 + 0.633i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.24T + 37T^{2} \)
41 \( 1 + (-3.86 - 6.69i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.328 - 0.568i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.48 + 2.56i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 1.03T + 53T^{2} \)
59 \( 1 + (-4.73 - 8.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.33 + 5.76i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.79 + 6.57i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.1T + 71T^{2} \)
73 \( 1 + 8.48T + 73T^{2} \)
79 \( 1 + (3.73 - 6.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.98 + 6.90i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 + (7.58 - 13.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.59078111525078118817168718065, −9.832245607011690877393334544884, −8.465909325133389243383149296861, −7.72562472702745139183128422896, −7.20273795405234882371162031456, −5.95320561005878114975776148756, −4.68306093664897330799884585128, −3.79464600876999890857437818036, −2.80613856604271623709203590468, −1.25675654337675002154113818526, 1.53242947596527587557465361742, 2.59421609424629520413451828112, 4.30126455076371145414959754040, 5.33897983340820475344637690316, 5.89293047716578530103074213125, 6.99072870948657576827946423838, 7.81095235386043055741995020698, 8.888620931440711908094111327880, 9.710410328302566048740849098956, 10.51671017141888690160241842675

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