Properties

Label 2-675-5.4-c3-0-64
Degree $2$
Conductor $675$
Sign $-0.894 + 0.447i$
Analytic cond. $39.8262$
Root an. cond. $6.31080$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.258i·2-s + 7.93·4-s − 14.5i·7-s + 4.12i·8-s − 49.2·11-s + 72.1i·13-s + 3.75·14-s + 62.3·16-s − 118. i·17-s − 123.·19-s − 12.7i·22-s − 91.4i·23-s − 18.6·26-s − 115. i·28-s − 174.·29-s + ⋯
L(s)  = 1  + 0.0914i·2-s + 0.991·4-s − 0.783i·7-s + 0.182i·8-s − 1.35·11-s + 1.53i·13-s + 0.0716·14-s + 0.974·16-s − 1.68i·17-s − 1.48·19-s − 0.123i·22-s − 0.829i·23-s − 0.140·26-s − 0.777i·28-s − 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(39.8262\)
Root analytic conductor: \(6.31080\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :3/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6212757489\)
\(L(\frac12)\) \(\approx\) \(0.6212757489\)
\(L(\frac{5}{2})\) 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 \( 1 - 0.258iT - 8T^{2} \)
7 \( 1 + 14.5iT - 343T^{2} \)
11 \( 1 + 49.2T + 1.33e3T^{2} \)
13 \( 1 - 72.1iT - 2.19e3T^{2} \)
17 \( 1 + 118. iT - 4.91e3T^{2} \)
19 \( 1 + 123.T + 6.85e3T^{2} \)
23 \( 1 + 91.4iT - 1.21e4T^{2} \)
29 \( 1 + 174.T + 2.43e4T^{2} \)
31 \( 1 + 46.2T + 2.97e4T^{2} \)
37 \( 1 + 154. iT - 5.06e4T^{2} \)
41 \( 1 + 364.T + 6.89e4T^{2} \)
43 \( 1 - 125. iT - 7.95e4T^{2} \)
47 \( 1 - 221. iT - 1.03e5T^{2} \)
53 \( 1 + 13.6iT - 1.48e5T^{2} \)
59 \( 1 + 239.T + 2.05e5T^{2} \)
61 \( 1 + 54.5T + 2.26e5T^{2} \)
67 \( 1 - 76.0iT - 3.00e5T^{2} \)
71 \( 1 - 728.T + 3.57e5T^{2} \)
73 \( 1 + 501. iT - 3.89e5T^{2} \)
79 \( 1 + 397.T + 4.93e5T^{2} \)
83 \( 1 + 1.36e3iT - 5.71e5T^{2} \)
89 \( 1 + 1.46e3T + 7.04e5T^{2} \)
97 \( 1 + 335. iT - 9.12e5T^{2} \)
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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

−9.883982318979785183456814034251, −8.839167851644644163532901635382, −7.72651242905089997586341670834, −7.09327112808991955639842608958, −6.38442931848689909853872108103, −5.15606432488061929765034508022, −4.15638810995570497326385646030, −2.77327484339063949094144368395, −1.87323775722988579917648050132, −0.14747141585450577505130855490, 1.76175452185414406853943400895, 2.66642465873837629777824048395, 3.68637661612565338784043519880, 5.41508969478809445375395560795, 5.81094373368700193800061468753, 6.93090089144567606082046519886, 8.090380598096620432930864017342, 8.352751852255948758536438133900, 9.896071298435939174177089975954, 10.61599319341061244287706251642

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