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Properties

Label 2-3675-105.83-c0-0-6
Degree $2$
Conductor $3675$
Sign $0.835 + 0.550i$
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  + (−0.707 + 0.707i)3-s − i·4-s − 1.00i·9-s + (0.707 + 0.707i)12-s + (−0.707 + 0.707i)13-s − 16-s + 1.73·19-s + (0.707 + 0.707i)27-s − 1.00·36-s + (1.22 − 1.22i)37-s − 1.00i·39-s + (0.707 − 0.707i)48-s + (0.707 + 0.707i)52-s + (−1.22 + 1.22i)57-s − 1.73i·61-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s − i·4-s − 1.00i·9-s + (0.707 + 0.707i)12-s + (−0.707 + 0.707i)13-s − 16-s + 1.73·19-s + (0.707 + 0.707i)27-s − 1.00·36-s + (1.22 − 1.22i)37-s − 1.00i·39-s + (0.707 − 0.707i)48-s + (0.707 + 0.707i)52-s + (−1.22 + 1.22i)57-s − 1.73i·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.835 + 0.550i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ 0.835 + 0.550i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9329337653\)
\(L(\frac12)\) \(\approx\) \(0.9329337653\)
\(L(1)\) 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 \( 1 + iT^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 - 1.73T + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.73iT - T^{2} \)
67 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
79 \( 1 + iT - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
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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

−9.051348363508889412813777303782, −7.79670305280133136964469840832, −6.97241847281746952863034206442, −6.28629242982728893160695359369, −5.51809503967332814530256741944, −4.98328032120985076462004291561, −4.27639170257888287469981041096, −3.24845043026351763888136030405, −1.99861722418109965796803947369, −0.73440946939596722750165792151, 1.05172536850922978426754093606, 2.46184614364847612438554450687, 3.10586528236397191837339102398, 4.25144028650659729618587787299, 5.12018165801596951838578453409, 5.75010038530105768961381294963, 6.79223219677790024643729156046, 7.30380821967754558853358626913, 7.915790519527754208051583689550, 8.462499547412289219475405280381

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