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Properties

Label 2-675-5.4-c3-0-47
Degree $2$
Conductor $675$
Sign $0.447 - 0.894i$
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  + 4.72i·2-s − 14.3·4-s + 17.6i·7-s − 29.9i·8-s + 34.2·11-s − 53.8i·13-s − 83.3·14-s + 26.9·16-s − 74.7i·17-s + 89.5·19-s + 161. i·22-s − 176. i·23-s + 254.·26-s − 252. i·28-s − 194.·29-s + ⋯
L(s)  = 1  + 1.67i·2-s − 1.79·4-s + 0.951i·7-s − 1.32i·8-s + 0.938·11-s − 1.14i·13-s − 1.59·14-s + 0.420·16-s − 1.06i·17-s + 1.08·19-s + 1.56i·22-s − 1.59i·23-s + 1.92·26-s − 1.70i·28-s − 1.24·29-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
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: \((2,\ 675,\ (\ :3/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.615501544\)
\(L(\frac12)\) \(\approx\) \(1.615501544\)
\(L(\frac{5}{2})\) 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 \( 1 - 4.72iT - 8T^{2} \)
7 \( 1 - 17.6iT - 343T^{2} \)
11 \( 1 - 34.2T + 1.33e3T^{2} \)
13 \( 1 + 53.8iT - 2.19e3T^{2} \)
17 \( 1 + 74.7iT - 4.91e3T^{2} \)
19 \( 1 - 89.5T + 6.85e3T^{2} \)
23 \( 1 + 176. iT - 1.21e4T^{2} \)
29 \( 1 + 194.T + 2.43e4T^{2} \)
31 \( 1 - 107.T + 2.97e4T^{2} \)
37 \( 1 + 430. iT - 5.06e4T^{2} \)
41 \( 1 + 108.T + 6.89e4T^{2} \)
43 \( 1 + 409. iT - 7.95e4T^{2} \)
47 \( 1 - 409. iT - 1.03e5T^{2} \)
53 \( 1 - 24.7iT - 1.48e5T^{2} \)
59 \( 1 - 295.T + 2.05e5T^{2} \)
61 \( 1 - 305.T + 2.26e5T^{2} \)
67 \( 1 - 915. iT - 3.00e5T^{2} \)
71 \( 1 - 228.T + 3.57e5T^{2} \)
73 \( 1 - 158. iT - 3.89e5T^{2} \)
79 \( 1 - 319.T + 4.93e5T^{2} \)
83 \( 1 - 936. iT - 5.71e5T^{2} \)
89 \( 1 + 920.T + 7.04e5T^{2} \)
97 \( 1 + 914. iT - 9.12e5T^{2} \)
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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

−9.793777218525966946865272842090, −9.059826275274318181205831242147, −8.433185225740946193135386596185, −7.46943339354265445160166596047, −6.78103668285957476586936976209, −5.69187978187702725561380286906, −5.32045787217768819356582321616, −4.06319475244828915353693545932, −2.60562890121891059890999962890, −0.53926782065493655482870811141, 1.12756400011564122002580853296, 1.78325951662628417579985383679, 3.41636113982084324442997689643, 3.90185685660007436088474209606, 4.90638325959927265424513382397, 6.39243419269281888849237557623, 7.39103635447807507299054738688, 8.583154429757997477454711383541, 9.596964696105042247489540823285, 9.890246438914951135119908481204

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