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Properties

Label 2-3675-3.2-c0-0-10
Degree $2$
Conductor $3675$
Sign $0.707 + 0.707i$
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·2-s + (0.707 + 0.707i)3-s + (0.707 − 0.707i)6-s − i·8-s + 1.00i·9-s − i·11-s + 1.41·13-s − 16-s + 1.41i·17-s + 1.00·18-s − 22-s + i·23-s + (0.707 − 0.707i)24-s − 1.41i·26-s + (−0.707 + 0.707i)27-s + ⋯
L(s)  = 1  − i·2-s + (0.707 + 0.707i)3-s + (0.707 − 0.707i)6-s − i·8-s + 1.00i·9-s − i·11-s + 1.41·13-s − 16-s + 1.41i·17-s + 1.00·18-s − 22-s + i·23-s + (0.707 − 0.707i)24-s − 1.41i·26-s + (−0.707 + 0.707i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.707 + 0.707i$
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: \((2,\ 3675,\ (\ :0),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.959330960\)
\(L(\frac12)\) \(\approx\) \(1.959330960\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + iT - T^{2} \)
11 \( 1 + iT - T^{2} \)
13 \( 1 - 1.41T + T^{2} \)
17 \( 1 - 1.41iT - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT - T^{2} \)
29 \( 1 + iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 1.41iT - T^{2} \)
61 \( 1 - 1.41T + T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 + iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
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

−8.700491365317502056316490788306, −8.211880358814604834133224188330, −7.31447245136424253652365745741, −6.15946058820900735427671602818, −5.71036625141730745396748180949, −4.31725028008344287913326208431, −3.69396393152958810004120085182, −3.25311686047738116299843225176, −2.20146919400396166781638149714, −1.29253921881875489189953796668, 1.33276758427688787673706172645, 2.38435083370556793039105663218, 3.16415103415593556032306943505, 4.34616811266516589480682689452, 5.20188624839065082997398439523, 6.16490160044925921214434116444, 6.78019234525748108316589941604, 7.16497917539555503939441779211, 8.064751837226306615658386770612, 8.445684813820667621404044548561

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