Properties

Label 2-3564-99.14-c0-0-1
Degree $2$
Conductor $3564$
Sign $-0.264 + 0.964i$
Analytic cond. $1.77866$
Root an. cond. $1.33366$
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  + (−0.251 − 0.564i)5-s + (−0.978 + 0.207i)7-s + (−0.406 + 0.913i)11-s + (0.0646 + 0.614i)13-s + (0.951 − 1.30i)17-s + (−0.309 − 0.951i)19-s + (−1.40 − 0.809i)23-s + (0.413 − 0.459i)25-s + (−0.207 − 0.978i)29-s + (0.104 + 0.994i)31-s + (0.363 + 0.5i)35-s + (0.207 − 0.978i)41-s + (−0.459 − 0.413i)47-s + (−0.587 − 0.809i)53-s + 0.618·55-s + ⋯
L(s)  = 1  + (−0.251 − 0.564i)5-s + (−0.978 + 0.207i)7-s + (−0.406 + 0.913i)11-s + (0.0646 + 0.614i)13-s + (0.951 − 1.30i)17-s + (−0.309 − 0.951i)19-s + (−1.40 − 0.809i)23-s + (0.413 − 0.459i)25-s + (−0.207 − 0.978i)29-s + (0.104 + 0.994i)31-s + (0.363 + 0.5i)35-s + (0.207 − 0.978i)41-s + (−0.459 − 0.413i)47-s + (−0.587 − 0.809i)53-s + 0.618·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.264 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.264 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3564\)    =    \(2^{2} \cdot 3^{4} \cdot 11\)
Sign: $-0.264 + 0.964i$
Analytic conductor: \(1.77866\)
Root analytic conductor: \(1.33366\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3564} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3564,\ (\ :0),\ -0.264 + 0.964i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6983511582\)
\(L(\frac12)\) \(\approx\) \(0.6983511582\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 + (0.406 - 0.913i)T \)
good5 \( 1 + (0.251 + 0.564i)T + (-0.669 + 0.743i)T^{2} \)
7 \( 1 + (0.978 - 0.207i)T + (0.913 - 0.406i)T^{2} \)
13 \( 1 + (-0.0646 - 0.614i)T + (-0.978 + 0.207i)T^{2} \)
17 \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (1.40 + 0.809i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.207 + 0.978i)T + (-0.913 + 0.406i)T^{2} \)
31 \( 1 + (-0.104 - 0.994i)T + (-0.978 + 0.207i)T^{2} \)
37 \( 1 + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (-0.207 + 0.978i)T + (-0.913 - 0.406i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.459 + 0.413i)T + (0.104 + 0.994i)T^{2} \)
53 \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.743 + 0.669i)T + (0.104 - 0.994i)T^{2} \)
61 \( 1 + (-0.0646 + 0.614i)T + (-0.978 - 0.207i)T^{2} \)
67 \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.669 + 0.743i)T^{2} \)
83 \( 1 + (0.994 + 0.104i)T + (0.978 + 0.207i)T^{2} \)
89 \( 1 - 0.618iT - T^{2} \)
97 \( 1 + (0.669 + 0.743i)T^{2} \)
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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

−8.501689517939896169602573424661, −7.85492276892073998225132807591, −6.89958989301875947809445872768, −6.50144713737852341002357597930, −5.37115313408915298843510304062, −4.74094718227274141660638435505, −3.94254624434104914621001296468, −2.88583425885892831361686830498, −2.06299866346539578329886669396, −0.40940974429919863161125871172, 1.39104301978937969607999016606, 2.82386808526844045173665886652, 3.49424097543787425096859332199, 4.00844106180454052885530992737, 5.52875127026947003376971982824, 5.93246250892488597825802522928, 6.61735122648634805948816625912, 7.71488121554915780218306679802, 7.985010739958505876279460124863, 8.889006721007841212264505881416

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