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Properties

Label 2-675-5.4-c3-0-41
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  + 4.24i·2-s − 9.99·4-s − 11i·7-s − 8.48i·8-s − 16.9·11-s + 29i·13-s + 46.6·14-s − 44.0·16-s − 50.9i·17-s − 29·19-s − 71.9i·22-s − 84.8i·23-s − 123.·26-s + 109. i·28-s + 271.·29-s + ⋯
L(s)  = 1  + 1.49i·2-s − 1.24·4-s − 0.593i·7-s − 0.374i·8-s − 0.465·11-s + 0.618i·13-s + 0.890·14-s − 0.687·16-s − 0.726i·17-s − 0.350·19-s − 0.697i·22-s − 0.769i·23-s − 0.928·26-s + 0.742i·28-s + 1.73·29-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

\(L(2)\) \(\approx\) \(1.314971240\)
\(L(\frac12)\) \(\approx\) \(1.314971240\)
\(L(\frac{5}{2})\) 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 \( 1 - 4.24iT - 8T^{2} \)
7 \( 1 + 11iT - 343T^{2} \)
11 \( 1 + 16.9T + 1.33e3T^{2} \)
13 \( 1 - 29iT - 2.19e3T^{2} \)
17 \( 1 + 50.9iT - 4.91e3T^{2} \)
19 \( 1 + 29T + 6.85e3T^{2} \)
23 \( 1 + 84.8iT - 1.21e4T^{2} \)
29 \( 1 - 271.T + 2.43e4T^{2} \)
31 \( 1 + 268T + 2.97e4T^{2} \)
37 \( 1 + 83iT - 5.06e4T^{2} \)
41 \( 1 - 271.T + 6.89e4T^{2} \)
43 \( 1 + 232iT - 7.95e4T^{2} \)
47 \( 1 + 390. iT - 1.03e5T^{2} \)
53 \( 1 + 305. iT - 1.48e5T^{2} \)
59 \( 1 - 288.T + 2.05e5T^{2} \)
61 \( 1 - 767T + 2.26e5T^{2} \)
67 \( 1 - 511iT - 3.00e5T^{2} \)
71 \( 1 + 712.T + 3.57e5T^{2} \)
73 \( 1 - 137iT - 3.89e5T^{2} \)
79 \( 1 - 475T + 4.93e5T^{2} \)
83 \( 1 + 576. iT - 5.71e5T^{2} \)
89 \( 1 + 254.T + 7.04e5T^{2} \)
97 \( 1 + 821iT - 9.12e5T^{2} \)
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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

−10.00489342129703294928296937011, −8.919670390492911808949329883193, −8.317020114713136794791375644755, −7.22258031394009241115553633789, −6.88666480581054054640523606807, −5.79029375426036735921512847601, −4.90953189922942182379449912882, −4.00755570109596809562192675923, −2.36252841239230965554550544667, −0.42524955904476876128093824758, 1.05609991305836804095847435330, 2.26696925705539302144033358066, 3.10787912137980774666463933827, 4.16079183228545776495765199034, 5.26925204592030912476725679032, 6.30696324771210892631215793799, 7.64421419211538119672552246132, 8.615300803233676026828180237019, 9.414072578296731999470509175503, 10.28048081896418504573795251312

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