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Properties

Label 2-675-5.4-c3-0-50
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  − 5.45i·2-s − 21.7·4-s − 11.8i·7-s + 75.3i·8-s + 56.2·11-s − 34.5i·13-s − 64.4·14-s + 236.·16-s − 39.2i·17-s + 146.·19-s − 306. i·22-s − 23.5i·23-s − 188.·26-s + 257. i·28-s + 161.·29-s + ⋯
L(s)  = 1  − 1.92i·2-s − 2.72·4-s − 0.637i·7-s + 3.32i·8-s + 1.54·11-s − 0.738i·13-s − 1.23·14-s + 3.69·16-s − 0.560i·17-s + 1.76·19-s − 2.97i·22-s − 0.213i·23-s − 1.42·26-s + 1.73i·28-s + 1.03·29-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

\(L(2)\) \(\approx\) \(1.760907920\)
\(L(\frac12)\) \(\approx\) \(1.760907920\)
\(L(\frac{5}{2})\) 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 \( 1 + 5.45iT - 8T^{2} \)
7 \( 1 + 11.8iT - 343T^{2} \)
11 \( 1 - 56.2T + 1.33e3T^{2} \)
13 \( 1 + 34.5iT - 2.19e3T^{2} \)
17 \( 1 + 39.2iT - 4.91e3T^{2} \)
19 \( 1 - 146.T + 6.85e3T^{2} \)
23 \( 1 + 23.5iT - 1.21e4T^{2} \)
29 \( 1 - 161.T + 2.43e4T^{2} \)
31 \( 1 + 29.5T + 2.97e4T^{2} \)
37 \( 1 + 217. iT - 5.06e4T^{2} \)
41 \( 1 + 142.T + 6.89e4T^{2} \)
43 \( 1 - 468. iT - 7.95e4T^{2} \)
47 \( 1 + 394. iT - 1.03e5T^{2} \)
53 \( 1 + 134. iT - 1.48e5T^{2} \)
59 \( 1 + 131.T + 2.05e5T^{2} \)
61 \( 1 - 259.T + 2.26e5T^{2} \)
67 \( 1 - 445. iT - 3.00e5T^{2} \)
71 \( 1 + 560.T + 3.57e5T^{2} \)
73 \( 1 - 88.6iT - 3.89e5T^{2} \)
79 \( 1 + 450.T + 4.93e5T^{2} \)
83 \( 1 - 284. iT - 5.71e5T^{2} \)
89 \( 1 - 625.T + 7.04e5T^{2} \)
97 \( 1 + 193. iT - 9.12e5T^{2} \)
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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

−9.800972122766313402551419780003, −9.197492694188904156078709148604, −8.257867400665622024029072627154, −7.09396363748346001480208877264, −5.56358803584879258377169468016, −4.53455339400145244915065487154, −3.65795828785714774806373067963, −2.86247161852678703589047662803, −1.39098051462690226431337754960, −0.65488685024632932822660433359, 1.18954853691406259459833781829, 3.51555513579795949921485424373, 4.49856808426133429779745627513, 5.45898359842495987393508351601, 6.29952195568298289647289675194, 6.93155931835507145886409776661, 7.82649769291461678439211073771, 8.858879074763650763569435823918, 9.189165704472642938266477181386, 10.08813421197130736597513562554

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