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Properties

Label 2-2925-65.34-c0-0-0
Degree $2$
Conductor $2925$
Sign $-0.168 - 0.985i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.168 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.168 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2925\)    =    \(3^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.168 - 0.985i$
Analytic conductor: \(1.45976\)
Root analytic conductor: \(1.20820\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2925} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2925,\ (\ :0),\ -0.168 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.262393412\)
\(L(\frac12)\) \(\approx\) \(1.262393412\)
\(L(1)\) 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

−8.877432391173805065918573551583, −8.452107150053014218798403494712, −7.63053142933933752874481358603, −7.19895575417817371103485691333, −6.03038001926911147283394911743, −5.17729934087253494757197033670, −4.57442749646993365564012442219, −3.47264431108151420695899848668, −2.63962268217671986742734840564, −1.73272705387019840550746620817, 0.813072678540964668806025214170, 1.83525056590084789969874532424, 2.94486511101192324381991584207, 4.33891591300115550590024760661, 4.81529581128304795641842446276, 5.49477803590909605609037465802, 6.55084536189553412616567618787, 7.23841852989063778615383911110, 7.83694491685249558842204458585, 8.796565446771411618030449092970

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