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Properties

Label 2-3675-105.62-c0-0-1
Degree $2$
Conductor $3675$
Sign $0.188 - 0.982i$
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  + (0.541 − 0.541i)2-s + (−0.707 + 0.707i)3-s + 0.414i·4-s + 0.765i·6-s + (0.765 + 0.765i)8-s − 1.00i·9-s + (−0.292 − 0.292i)12-s + 0.414·16-s + (1 + i)17-s + (−0.541 − 0.541i)18-s − 1.84·19-s + (1.30 + 1.30i)23-s − 1.08·24-s + (0.707 + 0.707i)27-s − 0.765i·31-s + (−0.541 + 0.541i)32-s + ⋯
L(s)  = 1  + (0.541 − 0.541i)2-s + (−0.707 + 0.707i)3-s + 0.414i·4-s + 0.765i·6-s + (0.765 + 0.765i)8-s − 1.00i·9-s + (−0.292 − 0.292i)12-s + 0.414·16-s + (1 + i)17-s + (−0.541 − 0.541i)18-s − 1.84·19-s + (1.30 + 1.30i)23-s − 1.08·24-s + (0.707 + 0.707i)27-s − 0.765i·31-s + (−0.541 + 0.541i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.188 - 0.982i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (2057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ 0.188 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.281876262\)
\(L(\frac12)\) \(\approx\) \(1.281876262\)
\(L(1)\) 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 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (-1 - i)T + iT^{2} \)
19 \( 1 + 1.84T + T^{2} \)
23 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + 0.765iT - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-1 - i)T + iT^{2} \)
53 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 0.765iT - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 + (1 - i)T - iT^{2} \)
89 \( 1 - T^{2} \)
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

−8.907073573150725479228039646913, −8.206815118395592123162237744596, −7.38940658162499334730159175799, −6.45641544809565393145759077769, −5.68139792970767762901414642550, −4.96992490109756488249418588342, −4.12536683897027322485150155341, −3.65958315956225542120858858005, −2.71794852732404034163631025463, −1.47731326695969124815660502240, 0.71248444538622746935575364005, 1.84911825278374920517032835188, 2.97096976835035211294744198774, 4.38976932380038960320514264833, 4.87155412483905414170207301199, 5.64454477724971907549909379044, 6.32855256979303276313989386132, 6.89531291986773811936952321966, 7.45669659854669901318258605398, 8.386107507838137879403022623024

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