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Properties

Label 2-3675-105.74-c0-0-9
Degree $2$
Conductor $3675$
Sign $-0.992 + 0.122i$
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  + (−0.5 − 0.866i)2-s + (0.965 − 0.258i)3-s + (−0.707 − 0.707i)6-s − 8-s + (0.866 − 0.499i)9-s + (−0.866 − 0.5i)11-s − 1.41i·13-s + (0.5 + 0.866i)16-s + (−0.707 + 1.22i)17-s + (−0.866 − 0.5i)18-s + 0.999i·22-s + (−0.5 − 0.866i)23-s + (−0.965 + 0.258i)24-s + (−1.22 + 0.707i)26-s + (0.707 − 0.707i)27-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.965 − 0.258i)3-s + (−0.707 − 0.707i)6-s − 8-s + (0.866 − 0.499i)9-s + (−0.866 − 0.5i)11-s − 1.41i·13-s + (0.5 + 0.866i)16-s + (−0.707 + 1.22i)17-s + (−0.866 − 0.5i)18-s + 0.999i·22-s + (−0.5 − 0.866i)23-s + (−0.965 + 0.258i)24-s + (−1.22 + 0.707i)26-s + (0.707 − 0.707i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-0.992 + 0.122i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (2174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ -0.992 + 0.122i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.107100508\)
\(L(\frac12)\) \(\approx\) \(1.107100508\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + 1.41iT - T^{2} \)
17 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + iT - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - iT - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
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

−8.532938758001575195802227240532, −7.972784162761310192817684453498, −7.04699024133092450816471285549, −6.09661851862929164953356637038, −5.48054869937406398530893668709, −4.14831642020703098730649016687, −3.33813754794851567005113545369, −2.55763881049289271673560503667, −1.93945323010125267684490845348, −0.61531571448270503956895222268, 1.86535922565505144281104896577, 2.70772812086220422238776770766, 3.56835048788202005722320990389, 4.59017079422324539550047051134, 5.26016398421004325251766986458, 6.50972699171485293123798129450, 7.04916028426963414469379609620, 7.60504579436333153981505024994, 8.269406463953715616312062446652, 9.004541780934573211349971417224

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