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Properties

Label 2-3675-3.2-c0-0-14
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  − i·3-s + 4-s − 9-s − i·12-s + 16-s − 2i·17-s + i·27-s − 36-s − 2i·47-s − i·48-s − 2·51-s + 64-s − 2i·68-s + 2·79-s + 81-s + ⋯
L(s)  = 1  − i·3-s + 4-s − 9-s − i·12-s + 16-s − 2i·17-s + i·27-s − 36-s − 2i·47-s − i·48-s − 2·51-s + 64-s − 2i·68-s + 2·79-s + 81-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

\(L(\frac{1}{2})\) \(\approx\) \(1.551270679\)
\(L(\frac12)\) \(\approx\) \(1.551270679\)
\(L(1)\) 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 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + 2iT - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + 2iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 2T + T^{2} \)
83 \( 1 - 2iT - T^{2} \)
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

−8.295839560881304784418916826783, −7.62755406240300367249423799818, −6.97965543151130922643185376026, −6.60180280615215932072809736414, −5.62226928676117078657477173185, −5.02620484761089965588994388323, −3.57395461599543331161586919306, −2.71488833962661591278711679466, −2.08178225693473971554350747416, −0.887688543538827763143135197367, 1.57122232089278006842471830638, 2.61476682433422393592787384883, 3.48892860635455666980821347254, 4.17428616714912115017015339593, 5.16004323093791286973895313532, 6.07541087600046416690785441334, 6.36261673405919710021221041910, 7.56040628296373223340304628115, 8.156989222462308298259129974444, 8.890085254511451302670749569358

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