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Properties

Label 2-3675-3.2-c0-0-8
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

\(L(\frac{1}{2})\) \(\approx\) \(1.219961022\)
\(L(\frac12)\) \(\approx\) \(1.219961022\)
\(L(1)\) 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

−8.780746628143600041479464935534, −8.186175518751827734609239188757, −7.02067788591528441656756870726, −6.14942841442118393600720440216, −5.11073602558918721954878506660, −4.51571736393598425069584229811, −3.68044538549805161037194842991, −2.99808214562133670811835669967, −2.25159638984118446757458155563, −0.822940456763734131056439174555, 1.18484180527247312075975038603, 2.44302775431214962710785042219, 3.43786074872914155191521368924, 4.73911736337185153559786748594, 5.50770477048211774475876571931, 6.01315399648944472947344957014, 6.80104535943236706382328453583, 7.41873162944595392880724940271, 7.83239285611836715104540234629, 8.565589603258524822273602893093

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