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Properties

Label 2-3264-408.101-c0-0-9
Degree $2$
Conductor $3264$
Sign $0.258 + 0.965i$
Analytic cond. $1.62894$
Root an. cond. $1.27630$
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·3-s − 1.73i·5-s − 9-s + i·11-s − 1.73i·13-s + 1.73·15-s + 17-s − i·19-s − 1.73·23-s − 1.99·25-s − i·27-s − 33-s + 1.73·39-s − 41-s − i·43-s + ⋯
L(s)  = 1  + i·3-s − 1.73i·5-s − 9-s + i·11-s − 1.73i·13-s + 1.73·15-s + 17-s − i·19-s − 1.73·23-s − 1.99·25-s − i·27-s − 33-s + 1.73·39-s − 41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3264\)    =    \(2^{6} \cdot 3 \cdot 17\)
Sign: $0.258 + 0.965i$
Analytic conductor: \(1.62894\)
Root analytic conductor: \(1.27630\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3264} (1121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3264,\ (\ :0),\ 0.258 + 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9810866803\)
\(L(\frac12)\) \(\approx\) \(0.9810866803\)
\(L(1)\) 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 \( 1 - iT \)
17 \( 1 - T \)
good5 \( 1 + 1.73iT - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - iT - T^{2} \)
13 \( 1 + 1.73iT - T^{2} \)
19 \( 1 + iT - T^{2} \)
23 \( 1 + 1.73T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + 2iT - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
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

−8.701211754624978364771878490885, −8.116799338303557392384019554625, −7.50518335530158334105500466828, −6.03922245919744545394966810682, −5.35722869105183291711686357844, −4.92202460233073808487471143664, −4.15119737514811157741690950143, −3.29960000589114888021866666765, −2.03672643876248542983809338444, −0.57846866493989287438974716546, 1.59817132773465775909791077745, 2.43561375941112266626914909062, 3.34478495687601401808718040363, 4.00636729930329663761055668099, 5.63771301072083091980415140643, 6.17986913264863319532358670495, 6.68493743347155157805162107260, 7.44986974490388435525083439309, 8.007759280248985988989656898860, 8.804390179707340998928802375423

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