Properties

Label 2-2475-5.4-c1-0-9
Degree $2$
Conductor $2475$
Sign $0.894 + 0.447i$
Analytic cond. $19.7629$
Root an. cond. $4.44555$
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  − 2.41i·2-s − 3.82·4-s + 0.414i·7-s + 4.41i·8-s + 11-s + 2.82i·13-s + 0.999·14-s + 2.99·16-s − 2.41i·17-s − 6.41·19-s − 2.41i·22-s + i·23-s + 6.82·26-s − 1.58i·28-s + 1.17·29-s + ⋯
L(s)  = 1  − 1.70i·2-s − 1.91·4-s + 0.156i·7-s + 1.56i·8-s + 0.301·11-s + 0.784i·13-s + 0.267·14-s + 0.749·16-s − 0.585i·17-s − 1.47·19-s − 0.514i·22-s + 0.208i·23-s + 1.33·26-s − 0.299i·28-s + 0.217·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(19.7629\)
Root analytic conductor: \(4.44555\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9732440559\)
\(L(\frac12)\) \(\approx\) \(0.9732440559\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 - T \)
good2 \( 1 + 2.41iT - 2T^{2} \)
7 \( 1 - 0.414iT - 7T^{2} \)
13 \( 1 - 2.82iT - 13T^{2} \)
17 \( 1 + 2.41iT - 17T^{2} \)
19 \( 1 + 6.41T + 19T^{2} \)
23 \( 1 - iT - 23T^{2} \)
29 \( 1 - 1.17T + 29T^{2} \)
31 \( 1 + 8.48T + 31T^{2} \)
37 \( 1 + 0.171iT - 37T^{2} \)
41 \( 1 - 10.8T + 41T^{2} \)
43 \( 1 - 11.6iT - 43T^{2} \)
47 \( 1 - 7.48iT - 47T^{2} \)
53 \( 1 - 7.65iT - 53T^{2} \)
59 \( 1 - 11T + 59T^{2} \)
61 \( 1 - 8.82T + 61T^{2} \)
67 \( 1 + 0.343iT - 67T^{2} \)
71 \( 1 + 7.82T + 71T^{2} \)
73 \( 1 - 8.82iT - 73T^{2} \)
79 \( 1 + 13.2T + 79T^{2} \)
83 \( 1 + 4.48iT - 83T^{2} \)
89 \( 1 - 3.65T + 89T^{2} \)
97 \( 1 - 5.82iT - 97T^{2} \)
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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

−9.164252935011949179778831236172, −8.551137181817035663919411156441, −7.45586358620511501456185291098, −6.51015314469844169360299720441, −5.52153634804185277992862589116, −4.36931673810985862742300796462, −4.05294231511408836867187843073, −2.85454320349412162939625092790, −2.15509539755876009654417942676, −1.12078926378051411794122827619, 0.36380217726064410013608527407, 2.17016719000257800325306429748, 3.74418375022745920690178854001, 4.34850907095944858481120054710, 5.47793647540764574041079792548, 5.83864675941068292794131522268, 6.85064998890396923875788243425, 7.22541549071877945912088325647, 8.273300086561140657440185602871, 8.569088326392266664379366328814

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