Properties

Label 2-3675-3.2-c0-0-16
Degree $2$
Conductor $3675$
Sign $-0.707 - 0.707i$
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  i·2-s + (−0.707 − 0.707i)3-s + (−0.707 + 0.707i)6-s i·8-s + 1.00i·9-s i·11-s − 1.41·13-s − 16-s − 1.41i·17-s + 1.00·18-s − 22-s + i·23-s + (−0.707 + 0.707i)24-s + 1.41i·26-s + (0.707 − 0.707i)27-s + ⋯
L(s)  = 1  i·2-s + (−0.707 − 0.707i)3-s + (−0.707 + 0.707i)6-s i·8-s + 1.00i·9-s i·11-s − 1.41·13-s − 16-s − 1.41i·17-s + 1.00·18-s − 22-s + i·23-s + (−0.707 + 0.707i)24-s + 1.41i·26-s + (0.707 − 0.707i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (1226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7106248893\)
\(L(\frac12)\) \(\approx\) \(0.7106248893\)
\(L(1)\) 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 \( 1 + iT - T^{2} \)
11 \( 1 + iT - T^{2} \)
13 \( 1 + 1.41T + T^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT - T^{2} \)
29 \( 1 + iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 1.41iT - T^{2} \)
61 \( 1 + 1.41T + T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 + iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
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

−8.015369036059407892295529323352, −7.52250099827301264934006335575, −6.75595507386167244135561715019, −6.07567536730872154551572410493, −5.17447730782899358320585416819, −4.44779369337215451099611065932, −3.13744964830412146197811810835, −2.58471975704953664346923979202, −1.56079329555129479106407873404, −0.42237920486310056561746350972, 1.82915343865953018499938799369, 2.91058576601281412184481586186, 4.25940603637756066494880695018, 4.73194048346737162737136349345, 5.54764394418439074861565651129, 6.16679320257199354842584322158, 6.99197151507144015431904743570, 7.37900778656576135678451678902, 8.392519086498711585717055601627, 9.040290746226829798432368861328

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