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Properties

Label 2-3264-8.5-c1-0-0
Degree $2$
Conductor $3264$
Sign $-0.258 + 0.965i$
Analytic cond. $26.0631$
Root an. cond. $5.10521$
Motivic weight $1$
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 + 3.92i·5-s + 0.343·7-s − 9-s + 0.853i·11-s − 4.26i·13-s − 3.92·15-s + 17-s + 0.492i·19-s + 0.343i·21-s − 7.38·23-s − 10.3·25-s − i·27-s − 10.2i·29-s − 5.17·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.75i·5-s + 0.129·7-s − 0.333·9-s + 0.257i·11-s − 1.18i·13-s − 1.01·15-s + 0.242·17-s + 0.113i·19-s + 0.0749i·21-s − 1.54·23-s − 2.07·25-s − 0.192i·27-s − 1.91i·29-s − 0.928·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3264\)    =    \(2^{6} \cdot 3 \cdot 17\)
Sign: $-0.258 + 0.965i$
Analytic conductor: \(26.0631\)
Root analytic conductor: \(5.10521\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3264} (1633, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3264,\ (\ :1/2),\ -0.258 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.04406360456\)
\(L(\frac12)\) \(\approx\) \(0.04406360456\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.92iT - 5T^{2} \)
7 \( 1 - 0.343T + 7T^{2} \)
11 \( 1 - 0.853iT - 11T^{2} \)
13 \( 1 + 4.26iT - 13T^{2} \)
19 \( 1 - 0.492iT - 19T^{2} \)
23 \( 1 + 7.38T + 23T^{2} \)
29 \( 1 + 10.2iT - 29T^{2} \)
31 \( 1 + 5.17T + 31T^{2} \)
37 \( 1 + 1.70iT - 37T^{2} \)
41 \( 1 + 2.85T + 41T^{2} \)
43 \( 1 + 4.49iT - 43T^{2} \)
47 \( 1 + 2.10T + 47T^{2} \)
53 \( 1 - 9.89iT - 53T^{2} \)
59 \( 1 - 9.58iT - 59T^{2} \)
61 \( 1 + 4.03iT - 61T^{2} \)
67 \( 1 + 10.6iT - 67T^{2} \)
71 \( 1 + 13.0T + 71T^{2} \)
73 \( 1 - 3.34T + 73T^{2} \)
79 \( 1 + 9.37T + 79T^{2} \)
83 \( 1 - 17.4iT - 83T^{2} \)
89 \( 1 - 10.2T + 89T^{2} \)
97 \( 1 + 10.2T + 97T^{2} \)
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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

−9.449794640378249412175147012998, −8.172471724651500605100290921105, −7.75130594295548871363975722029, −6.97226669433417542030726890651, −6.00955606173097606985296721623, −5.66766572499509081032836237980, −4.35661141092196837313948162656, −3.61170671256254966910785630010, −2.87457378784037201964514024953, −2.04458260540151404873687389515, 0.01276102422204978274309105837, 1.38764010303526375422486229066, 1.88792794781643608125570613161, 3.42643461698542148841522432790, 4.36411557112516501199981618685, 5.05002040495406483877809781145, 5.76222212352045310852537551887, 6.61604130188641432471879084137, 7.47687825167406225058119135836, 8.307031018176277582893110861029

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