Properties

Label 2-2925-5.4-c1-0-27
Degree $2$
Conductor $2925$
Sign $-0.894 + 0.447i$
Analytic cond. $23.3562$
Root an. cond. $4.83282$
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  + 2i·2-s − 2·4-s + 3i·7-s + 5·11-s + i·13-s − 6·14-s − 4·16-s + 5i·17-s − 2·19-s + 10i·22-s + i·23-s − 2·26-s − 6i·28-s + 10·29-s − 2·31-s − 8i·32-s + ⋯
L(s)  = 1  + 1.41i·2-s − 4-s + 1.13i·7-s + 1.50·11-s + 0.277i·13-s − 1.60·14-s − 16-s + 1.21i·17-s − 0.458·19-s + 2.13i·22-s + 0.208i·23-s − 0.392·26-s − 1.13i·28-s + 1.85·29-s − 0.359·31-s − 1.41i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2925\)    =    \(3^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(23.3562\)
Root analytic conductor: \(4.83282\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2925} (2224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2925,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.810665242\)
\(L(\frac12)\) \(\approx\) \(1.810665242\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 - iT \)
good2 \( 1 - 2iT - 2T^{2} \)
7 \( 1 - 3iT - 7T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
17 \( 1 - 5iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - iT - 23T^{2} \)
29 \( 1 - 10T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 - 3iT - 37T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 10iT - 47T^{2} \)
53 \( 1 + 9iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 11T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 15T + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 - 11T + 79T^{2} \)
83 \( 1 + 8iT - 83T^{2} \)
89 \( 1 + 11T + 89T^{2} \)
97 \( 1 - 9iT - 97T^{2} \)
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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

−8.884456166963231014604933930854, −8.452786135679513960131907016705, −7.66492130684733042809321401829, −6.66144753550010207973464178678, −6.27135319440274831991477037726, −5.71290059337612206925382099007, −4.69536624586676876610615779475, −3.99007128453281432375083029977, −2.66787646879889425397448257677, −1.53472755317651973535059309969, 0.62469013096305254382519254061, 1.34851762432787253956217612460, 2.56204718516906521627184556147, 3.40357829694951989747006121101, 4.22748431481126597095443123280, 4.67030000607447335102972201670, 6.12942240015596643193643312437, 6.88118108496805208111348224235, 7.49686593032851406998557892223, 8.675566139624296525773197682312

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