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Properties

Label 2-3675-3.2-c0-0-15
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

\(L(\frac{1}{2})\) \(\approx\) \(1.931593908\)
\(L(\frac12)\) \(\approx\) \(1.931593908\)
\(L(1)\) 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

−8.716649891644985363828372708510, −7.40267347428367918236611533933, −7.26380921635029310835804218598, −6.39592748598372648674808152101, −5.42797162925783010994348969792, −4.19995031645120723927880226800, −3.48269768996560425530763397016, −2.76012933716369695807839437244, −1.85806899579098128342817593900, −1.11559764815746007239669953410, 1.67683023838502856194614640681, 2.79638840360107294914802685752, 3.65720165423520020552313369217, 4.40730992175824441496599941227, 5.43119381009745393309045021971, 6.13393537860390401221895768211, 6.56867759499328657040981595458, 7.81540918724401006027645799992, 8.250410589010700508815022039964, 8.632613900910304660989904180291

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