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Properties

Label 2-3675-105.47-c0-0-1
Degree $2$
Conductor $3675$
Sign $-0.358 - 0.933i$
Analytic cond. $1.83406$
Root an. cond. $1.35427$
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  + (0.198 + 0.739i)2-s + (−0.965 − 0.258i)3-s + (0.358 − 0.207i)4-s − 0.765i·6-s + (0.765 + 0.765i)8-s + (0.866 + 0.499i)9-s + (−0.400 + 0.107i)12-s + (−0.207 + 0.358i)16-s + (−0.366 + 1.36i)17-s + (−0.198 + 0.739i)18-s + (−0.923 + 1.60i)19-s + (−1.78 + 0.478i)23-s + (−0.541 − 0.937i)24-s + (−0.707 − 0.707i)27-s + (0.662 − 0.382i)31-s + (0.739 + 0.198i)32-s + ⋯
L(s)  = 1  + (0.198 + 0.739i)2-s + (−0.965 − 0.258i)3-s + (0.358 − 0.207i)4-s − 0.765i·6-s + (0.765 + 0.765i)8-s + (0.866 + 0.499i)9-s + (−0.400 + 0.107i)12-s + (−0.207 + 0.358i)16-s + (−0.366 + 1.36i)17-s + (−0.198 + 0.739i)18-s + (−0.923 + 1.60i)19-s + (−1.78 + 0.478i)23-s + (−0.541 − 0.937i)24-s + (−0.707 − 0.707i)27-s + (0.662 − 0.382i)31-s + (0.739 + 0.198i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-0.358 - 0.933i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (1832, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ -0.358 - 0.933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.042300236\)
\(L(\frac12)\) \(\approx\) \(1.042300236\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.198 - 0.739i)T + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (1.78 - 0.478i)T + (0.866 - 0.5i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-0.662 + 0.382i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.866 - 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (0.478 - 1.78i)T + (-0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.662 - 0.382i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.866 - 0.5i)T^{2} \)
79 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-1 + i)T - iT^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - iT^{2} \)
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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

−8.602879627065424953946509993931, −7.914989487494485731506586101780, −7.42777173051853972595493429282, −6.39015925625122785053007120224, −6.03579122449772525557495383675, −5.62563575481204549376920434277, −4.45190581145557725682327621620, −3.93100820005409147797082946478, −2.18199424594700963073564694556, −1.52778079010485811923395335398, 0.61801979358314675053550827620, 2.04377664561270429016014815511, 2.81066156236963895270674140342, 3.97229076922932782168646677568, 4.52587184844491840662920727419, 5.29872558995285782989522072435, 6.44087623927901746820159962238, 6.77526167088313628110704810453, 7.57790442526327983488009711415, 8.559016512683444526213492656394

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