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Properties

Label 2-2925-5.4-c1-0-14
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  + i·2-s + 4-s + 4i·7-s + 3i·8-s − 4·11-s + i·13-s − 4·14-s − 16-s + 2i·17-s − 4i·22-s − 26-s + 4i·28-s − 10·29-s + 4·31-s + 5i·32-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.5·4-s + 1.51i·7-s + 1.06i·8-s − 1.20·11-s + 0.277i·13-s − 1.06·14-s − 0.250·16-s + 0.485i·17-s − 0.852i·22-s − 0.196·26-s + 0.755i·28-s − 1.85·29-s + 0.718·31-s + 0.883i·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

\(L(1)\) \(\approx\) \(1.207454890\)
\(L(\frac12)\) \(\approx\) \(1.207454890\)
\(L(\frac{3}{2})\) 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 \( 1 - iT - 2T^{2} \)
7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 10T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 12iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−8.772690652692617962106754501795, −8.488049936849601662791879763476, −7.63295805216301077889711192275, −6.95199897208198663300635311659, −6.02446013151447915170303196880, −5.52897037324362149421164876483, −4.96285408740251922171154036978, −3.52300394618510796945620991644, −2.46681253267662140395938145967, −1.95573353710866215063471080347, 0.34372289651979761721810109214, 1.45075543654873564354733847921, 2.57855157966598001635863159004, 3.37538769670545714128455133228, 4.17087109576072520398871798955, 5.09618066963863900792939241798, 6.09488652561611503424574208184, 7.01704765163057923983705989247, 7.52215925676824301498297475112, 8.105137708122562591303314349168

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