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Properties

Label 2-2904-264.131-c0-0-8
Degree $2$
Conductor $2904$
Sign $0.426 - 0.904i$
Analytic cond. $1.44928$
Root an. cond. $1.20386$
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 + 3-s − 4-s + 1.41·5-s + i·6-s − i·8-s + 9-s + 1.41i·10-s − 12-s + 1.41·15-s + 16-s + i·18-s − 1.41i·19-s − 1.41·20-s − 1.41·23-s − i·24-s + ⋯
L(s)  = 1  + i·2-s + 3-s − 4-s + 1.41·5-s + i·6-s − i·8-s + 9-s + 1.41i·10-s − 12-s + 1.41·15-s + 16-s + i·18-s − 1.41i·19-s − 1.41·20-s − 1.41·23-s − i·24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2904\)    =    \(2^{3} \cdot 3 \cdot 11^{2}\)
Sign: $0.426 - 0.904i$
Analytic conductor: \(1.44928\)
Root analytic conductor: \(1.20386\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2904} (1451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2904,\ (\ :0),\ 0.426 - 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.027205115\)
\(L(\frac12)\) \(\approx\) \(2.027205115\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 - T \)
11 \( 1 \)
good5 \( 1 - 1.41T + T^{2} \)
7 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + 1.41iT - T^{2} \)
23 \( 1 + 1.41T + T^{2} \)
29 \( 1 - 2iT - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + 1.41iT - T^{2} \)
47 \( 1 - 1.41T + T^{2} \)
53 \( 1 + 1.41T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 - 1.41iT - T^{2} \)
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

−8.931251298092524198008041998781, −8.505815651060815911002542487493, −7.46039961359416901731727542061, −6.89895369193751684129599225614, −6.14362671429121904547827014088, −5.31568013000276066451791287837, −4.58288464312716362579777842917, −3.54211014362887163916126462815, −2.52217068175986871483678873913, −1.47545078631259128152162531697, 1.51933748378733798441512312136, 2.09517407449437172558319255002, 2.91527372999168091364916218457, 3.90219929951433165132950187356, 4.61841461382272107777022731172, 5.80908791945474832130841626727, 6.24961475351143979367112744422, 7.77828044463666862838756292339, 8.121896680354990229483162362035, 9.120538915063700493544328407604

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