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Properties

Label 2-2912-728.237-c0-0-3
Degree $2$
Conductor $2912$
Sign $-0.964 + 0.265i$
Analytic cond. $1.45327$
Root an. cond. $1.20551$
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  + (−0.5 − 0.866i)3-s − 5-s + (0.5 − 0.866i)7-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)15-s + (1 − 1.73i)19-s − 0.999·21-s + (−0.5 − 0.866i)23-s − 27-s + (−0.5 + 0.866i)35-s + 0.999·39-s + (−0.499 − 0.866i)49-s − 1.99·57-s + (−0.5 + 0.866i)59-s + (0.5 − 0.866i)61-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s − 5-s + (0.5 − 0.866i)7-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)15-s + (1 − 1.73i)19-s − 0.999·21-s + (−0.5 − 0.866i)23-s − 27-s + (−0.5 + 0.866i)35-s + 0.999·39-s + (−0.499 − 0.866i)49-s − 1.99·57-s + (−0.5 + 0.866i)59-s + (0.5 − 0.866i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2912\)    =    \(2^{5} \cdot 7 \cdot 13\)
Sign: $-0.964 + 0.265i$
Analytic conductor: \(1.45327\)
Root analytic conductor: \(1.20551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2912} (1329, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2912,\ (\ :0),\ -0.964 + 0.265i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6106815354\)
\(L(\frac12)\) \(\approx\) \(0.6106815354\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + T + T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 2T + T^{2} \)
83 \( 1 + 2T + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−8.392321911832344609577633383455, −7.60735608985001073705742734953, −7.07790622551070176670164476830, −6.71176405334994841082024049733, −5.54533193705122803380408472379, −4.52082181158414397346474637062, −4.07882246868866358876862425681, −2.83891088804203553756896529860, −1.55178590512465365259516325780, −0.42222309126660199384717566941, 1.67830505357173914396711680693, 3.05114283979726726854812904867, 3.86506417585201058261209289149, 4.62076945894747996870930458912, 5.57090629017642053251071850377, 5.74314054025628627869151192986, 7.28383110894695504202636867993, 7.889848428297629865573737666616, 8.336661709220891967036241552878, 9.494911706228323779794393358226

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