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Properties

Label 2-3675-3.2-c0-0-0
Degree $2$
Conductor $3675$
Sign $i$
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  + 1.41i·2-s + i·3-s − 1.00·4-s − 1.41·6-s − 9-s − 1.00i·12-s − 0.999·16-s − 1.41i·18-s − 1.41·19-s + 1.41i·23-s − i·27-s − 1.41·31-s − 1.41i·32-s + 1.00·36-s − 2.00i·38-s + ⋯
L(s)  = 1  + 1.41i·2-s + i·3-s − 1.00·4-s − 1.41·6-s − 9-s − 1.00i·12-s − 0.999·16-s − 1.41i·18-s − 1.41·19-s + 1.41i·23-s − i·27-s − 1.41·31-s − 1.41i·32-s + 1.00·36-s − 2.00i·38-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (1226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7002867323\)
\(L(\frac12)\) \(\approx\) \(0.7002867323\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 - 1.41iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 1.41T + T^{2} \)
23 \( 1 - 1.41iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.41T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.41T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.069788206621324835387514032581, −8.509252842853381353741081116711, −7.73127079211942958824486123664, −7.08881193823097132927229411213, −6.09560513501837712300660610237, −5.71697972405204197144825293321, −4.86299519466543540237280624927, −4.20988410176231920015862645776, −3.29225496567429635124775422546, −2.06566702077787859214345016524, 0.37246897971375030824601561153, 1.69013764664760250571454696779, 2.28217696657116332585095021222, 3.13190296037312137611406742292, 4.03948349753073161448589278225, 4.91941276626149735131528971085, 6.05327040200752239837873592710, 6.67381692758434237713266800860, 7.41281092431860943742633016264, 8.426103545618263512352288365845

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