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Properties

Label 2-2925-13.8-c0-0-0
Degree $2$
Conductor $2925$
Sign $-0.289 + 0.957i$
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 − i·13-s − 16-s + (−1 − i)19-s + (−1 − i)28-s + (1 + i)31-s + (−1 + i)37-s − i·49-s − 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 − i·13-s − 16-s + (−1 − i)19-s + (−1 − i)28-s + (1 + i)31-s + (−1 + i)37-s − i·49-s − 52-s + i·64-s + (1 + i)67-s + (1 − i)73-s + (−1 + i)76-s + (−1 − i)91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2925\)    =    \(3^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.289 + 0.957i$
Analytic conductor: \(1.45976\)
Root analytic conductor: \(1.20820\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2925} (1126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2925,\ (\ :0),\ -0.289 + 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.256228095\)
\(L(\frac12)\) \(\approx\) \(1.256228095\)
\(L(1)\) 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 \( 1 + iT \)
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

−8.578463707715024107279476466777, −8.096598058912461166318883789972, −7.05267160860198830843268447143, −6.56959945734339281681644154544, −5.49558565105280841295918485178, −4.85374351795357824825049766707, −4.24394926466181307307903669687, −2.96332388599962322690076830622, −1.78382365759962944775249701589, −0.797753375799117961488007663994, 1.87717345643110840097775915967, 2.45039883942702676141526581431, 3.72720249732470050320271435260, 4.37794120111384212584934433152, 5.24270426378048592321858880200, 6.18498535589391991572459290462, 6.94351350092379117722643286356, 7.924785992434291661239084190691, 8.321117685088286588720852072272, 8.961648947526123855276460415419

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