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Properties

Label 2-3675-105.38-c0-0-2
Degree $2$
Conductor $3675$
Sign $-0.135 + 0.990i$
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  + (0.478 − 1.78i)2-s + (−0.965 + 0.258i)3-s + (−2.09 − 1.20i)4-s + 1.84i·6-s + (−1.84 + 1.84i)8-s + (0.866 − 0.499i)9-s + (2.33 + 0.624i)12-s + (1.20 + 2.09i)16-s + (0.366 + 1.36i)17-s + (−0.478 − 1.78i)18-s + (0.382 + 0.662i)19-s + (0.739 + 0.198i)23-s + (1.30 − 2.26i)24-s + (−0.707 + 0.707i)27-s + (1.60 + 0.923i)31-s + (1.78 − 0.478i)32-s + ⋯
L(s)  = 1  + (0.478 − 1.78i)2-s + (−0.965 + 0.258i)3-s + (−2.09 − 1.20i)4-s + 1.84i·6-s + (−1.84 + 1.84i)8-s + (0.866 − 0.499i)9-s + (2.33 + 0.624i)12-s + (1.20 + 2.09i)16-s + (0.366 + 1.36i)17-s + (−0.478 − 1.78i)18-s + (0.382 + 0.662i)19-s + (0.739 + 0.198i)23-s + (1.30 − 2.26i)24-s + (−0.707 + 0.707i)27-s + (1.60 + 0.923i)31-s + (1.78 − 0.478i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-0.135 + 0.990i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (668, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ -0.135 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.010677213\)
\(L(\frac12)\) \(\approx\) \(1.010677213\)
\(L(1)\) 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 \( 1 + (-0.478 + 1.78i)T + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.739 - 0.198i)T + (0.866 + 0.5i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-1.60 - 0.923i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.866 + 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.198 - 0.739i)T + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-1.60 + 0.923i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.866 + 0.5i)T^{2} \)
79 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (1 + i)T + iT^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + iT^{2} \)
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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

−8.826968842273735017408595624752, −8.013087460629541543849000036022, −6.74080282991763608368087134789, −5.93611106462336639569337110647, −5.20374418799664210803288397427, −4.56946132601278634510347474893, −3.75373987252816986682323858846, −3.12200287255179692546606459015, −1.81776866418598513947407461539, −1.02361241859950335335140153982, 0.76342559728662859154927675580, 2.75851606948940038763315554312, 3.99147771527886147658604805332, 4.89337127957755022132174430949, 5.15498486272792545432526553329, 6.02890785746652707401327835873, 6.71618681811696928176384848504, 7.15449983613721685418753051335, 7.84971096612991454373525374261, 8.555942931005062814574591740638

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