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Properties

Label 2-3675-105.38-c0-0-3
Degree $2$
Conductor $3675$
Sign $0.964 - 0.265i$
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  + (−0.258 − 0.965i)3-s + (0.866 + 0.5i)4-s + (−0.866 + 0.499i)9-s + (0.258 − 0.965i)12-s + (−0.707 + 0.707i)13-s + (0.499 + 0.866i)16-s + (0.866 + 1.5i)19-s + (0.707 + 0.707i)27-s − 36-s + (1.67 + 0.448i)37-s + (0.866 + 0.500i)39-s + (0.707 − 0.707i)48-s + (−0.965 + 0.258i)52-s + (1.22 − 1.22i)57-s + (1.5 − 0.866i)61-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)3-s + (0.866 + 0.5i)4-s + (−0.866 + 0.499i)9-s + (0.258 − 0.965i)12-s + (−0.707 + 0.707i)13-s + (0.499 + 0.866i)16-s + (0.866 + 1.5i)19-s + (0.707 + 0.707i)27-s − 36-s + (1.67 + 0.448i)37-s + (0.866 + 0.500i)39-s + (0.707 − 0.707i)48-s + (−0.965 + 0.258i)52-s + (1.22 − 1.22i)57-s + (1.5 − 0.866i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.964 - 0.265i$
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: \((2,\ 3675,\ (\ :0),\ 0.964 - 0.265i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.365650021\)
\(L(\frac12)\) \(\approx\) \(1.365650021\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
17 \( 1 + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-1.67 - 0.448i)T + (0.866 + 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.448 + 1.67i)T + (-0.866 + 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
79 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
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

−8.389512391533676639048684307553, −7.84323779980519326834928406022, −7.30991069367307104986968636107, −6.58853240048017183351073043645, −5.99865572607450727686357744724, −5.17342539778882892054271450900, −4.00516147917412917173023270736, −3.01829352135375402252832802218, −2.20478517182560553954634276663, −1.35661043091872302330523056939, 0.850481352622514594211939691985, 2.50954305567291812728206334491, 2.98572472125017055766951590534, 4.15644170311274365915882302009, 5.07394034850124022629995416061, 5.53555184935899938957843145000, 6.35874893074043774523828796786, 7.16707221096268343070935824070, 7.83149142505741159829350521279, 8.896142316260864274675022333487

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