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Properties

Label 2-3675-105.17-c0-0-3
Degree $2$
Conductor $3675$
Sign $0.922 + 0.385i$
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  + (−0.965 + 0.258i)3-s + (−0.866 − 0.5i)4-s + (0.866 − 0.499i)9-s + (0.965 + 0.258i)12-s + (0.707 + 0.707i)13-s + (0.499 + 0.866i)16-s + (−0.866 − 1.5i)19-s + (−0.707 + 0.707i)27-s − 36-s + (−0.448 + 1.67i)37-s + (−0.866 − 0.500i)39-s + (−0.707 − 0.707i)48-s + (−0.258 − 0.965i)52-s + (1.22 + 1.22i)57-s + (1.5 − 0.866i)61-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)3-s + (−0.866 − 0.5i)4-s + (0.866 − 0.499i)9-s + (0.965 + 0.258i)12-s + (0.707 + 0.707i)13-s + (0.499 + 0.866i)16-s + (−0.866 − 1.5i)19-s + (−0.707 + 0.707i)27-s − 36-s + (−0.448 + 1.67i)37-s + (−0.866 − 0.500i)39-s + (−0.707 − 0.707i)48-s + (−0.258 − 0.965i)52-s + (1.22 + 1.22i)57-s + (1.5 − 0.866i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.922 + 0.385i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (2432, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ 0.922 + 0.385i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6921426928\)
\(L(\frac12)\) \(\approx\) \(0.6921426928\)
\(L(1)\) 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 \( 1 + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
17 \( 1 + (0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
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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

−8.812341544505657164869732993522, −8.094096400282238378094985362225, −6.73986589130961812055589847927, −6.54073054508696431030262245623, −5.55064181541332301803340673875, −4.85215911155286880652153373418, −4.32979611693467993705425677401, −3.45532558182185095089451475552, −1.88784294735161517034862152344, −0.69951722517531344810381099825, 0.846542425733364151394629196099, 2.16668199259341961826649755866, 3.64871896448099961313709986396, 4.05601054249385770119369013870, 5.14685568656688517958600273029, 5.66409435115214759723070922478, 6.42729950473977447304276976674, 7.34994480537380617049862077675, 8.056959363510875456466744630956, 8.574844407386863013690844226227

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