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Properties

Label 2-3675-105.83-c0-0-10
Degree $2$
Conductor $3675$
Sign $-0.372 + 0.928i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.372 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.372 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-0.372 + 0.928i$
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: \((2,\ 3675,\ (\ :0),\ -0.372 + 0.928i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.677582227\)
\(L(\frac12)\) \(\approx\) \(1.677582227\)
\(L(1)\) 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

−8.527471153640726240759696176712, −7.71360975825310691863963827745, −7.04018787610996090175320891294, −6.27036691676573558300984481325, −5.60514997354153840024786722614, −4.82335329138317175769783747360, −3.58273384009855928172905999792, −2.90027751741105085253290193822, −1.72514242960772954320825151776, −0.951498143405278233485002290026, 1.74152286705688323430625370125, 2.84194590857153043208938121814, 3.49876983008781627087225229636, 4.13911453382389246755624348480, 4.96861184616286059679763050150, 5.87322387002971287048230624797, 7.07313867179011044093641800422, 7.50682019659833403308883675071, 8.306773095666319211142742454011, 8.983516114950585768579679868360

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