Properties

Label 2-2925-5.4-c1-0-23
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s − 2·4-s + 3i·7-s + 11-s + i·13-s + 6·14-s − 4·16-s + i·17-s + 2·19-s − 2i·22-s − 3i·23-s + 2·26-s − 6i·28-s − 2·29-s − 6·31-s + 8i·32-s + ⋯
L(s)  = 1  − 1.41i·2-s − 4-s + 1.13i·7-s + 0.301·11-s + 0.277i·13-s + 1.60·14-s − 16-s + 0.242i·17-s + 0.458·19-s − 0.426i·22-s − 0.625i·23-s + 0.392·26-s − 1.13i·28-s − 0.371·29-s − 1.07·31-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

\(L(1)\) \(\approx\) \(1.549107845\)
\(L(\frac12)\) \(\approx\) \(1.549107845\)
\(L(\frac{3}{2})\) 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 \( 1 - T + 11T^{2} \)
17 \( 1 - iT - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + 3iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 - 11iT - 37T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 10iT - 47T^{2} \)
53 \( 1 - 11iT - 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 - 13T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 - 5T + 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 - 3T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
97 \( 1 - 17iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.037008400611670573419571767770, −8.269985654050410196391348434937, −7.19744131751515319901032682505, −6.32810379195930940007664517600, −5.50068394414761151734445971764, −4.55577859397052027099335077742, −3.71648278380554459268328947114, −2.81377589737374436834965191329, −2.13667853204613269870672906780, −1.11616088887264556073464043862, 0.54684343669856144119168045931, 2.06356156990502496368370980074, 3.55103952631971567866707456847, 4.20855102704911379404082232144, 5.31213003662936222885185532060, 5.70001926955390453610685032465, 6.92908566360159277454193116003, 7.07978112278912470948974670538, 7.84435811853760561171427918230, 8.538322671737764831684737668083

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