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Properties

Label 2-2925-65.44-c0-0-0
Degree $2$
Conductor $2925$
Sign $0.687 + 0.726i$
Analytic cond. $1.45976$
Root an. cond. $1.20820$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·4-s + (−1 + i)7-s + 13-s − 16-s + (1 − i)19-s + (1 + i)28-s + (1 − i)31-s + (1 − i)37-s − i·49-s − i·52-s + i·64-s + (1 + i)67-s + (1 − i)73-s + (−1 − i)76-s + (−1 + i)91-s + ⋯
L(s)  = 1  − i·4-s + (−1 + i)7-s + 13-s − 16-s + (1 − i)19-s + (1 + i)28-s + (1 − i)31-s + (1 − i)37-s − i·49-s − i·52-s + i·64-s + (1 + i)67-s + (1 − i)73-s + (−1 − i)76-s + (−1 + i)91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2925\)    =    \(3^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.687 + 0.726i$
Analytic conductor: \(1.45976\)
Root analytic conductor: \(1.20820\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2925} (2449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2925,\ (\ :0),\ 0.687 + 0.726i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.134353950\)
\(L(\frac12)\) \(\approx\) \(1.134353950\)
\(L(1)\) 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 \( 1 - T \)
good2 \( 1 + iT^{2} \)
7 \( 1 + (1 - i)T - iT^{2} \)
11 \( 1 + iT^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (-1 + i)T - iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-1 + i)T - iT^{2} \)
37 \( 1 + (-1 + i)T - iT^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (-1 - i)T + iT^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (-1 + i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (1 + i)T + iT^{2} \)
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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

−9.097407592431904067470331168552, −8.249313731272776710654606122550, −7.15485647259275612211134353752, −6.34361901833019919199888573346, −5.88636541586108532440335442379, −5.18744245549190367212731792590, −4.16427502390397744213224719821, −3.04201429419698191488151207029, −2.24686146542404139963555246619, −0.859533339486763757477469956919, 1.17424830026051623143488192957, 2.78208793556568982001997120457, 3.54135330926877451265211800089, 4.00413991492576385201963831895, 5.08973662254021009056302118022, 6.32056528747818534711733054607, 6.70561597173879844154223885900, 7.65393889534254582285729320616, 8.129456183186555261499199370028, 8.985993131654618376910504969253

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