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Properties

Label 2-2912-364.51-c0-0-5
Degree $2$
Conductor $2912$
Sign $-0.197 + 0.980i$
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  + (−0.866 + 0.5i)3-s + (0.866 + 0.5i)5-s − 7-s + (−0.5 − 0.866i)11-s − i·13-s − 0.999·15-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.866 − 0.5i)21-s + (0.866 + 0.5i)23-s − i·27-s − 2·29-s + (−0.5 − 0.866i)31-s + (0.866 + 0.499i)33-s + (−0.866 − 0.5i)35-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (0.866 + 0.5i)5-s − 7-s + (−0.5 − 0.866i)11-s − i·13-s − 0.999·15-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.866 − 0.5i)21-s + (0.866 + 0.5i)23-s − i·27-s − 2·29-s + (−0.5 − 0.866i)31-s + (0.866 + 0.499i)33-s + (−0.866 − 0.5i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2912\)    =    \(2^{5} \cdot 7 \cdot 13\)
Sign: $-0.197 + 0.980i$
Analytic conductor: \(1.45327\)
Root analytic conductor: \(1.20551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2912} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2912,\ (\ :0),\ -0.197 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3488982246\)
\(L(\frac12)\) \(\approx\) \(0.3488982246\)
\(L(1)\) 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 \( 1 + T \)
13 \( 1 + iT \)
good3 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + 2T + T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
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

−8.990119941322849515940564298206, −7.88834741391844893581569521344, −7.10072183825322856261424391943, −6.13132281315134269402456213948, −5.69024094068057858222615895968, −5.24481603004707322515779495712, −3.91134631304932155851177665696, −3.07055693500595947810915855912, −2.18081425909681132959020909403, −0.22976991926940190060396467814, 1.48776760909397146447648582571, 2.34516782241220873097103135638, 3.63962184067883071937350991156, 4.70675071485074325148338161944, 5.45874583482999725797307134991, 6.14790080510796345298794301244, 6.85239806231753941012932892978, 7.21750428414303577109340021964, 8.671381296291605357173894482746, 9.213512185208611898340578314129

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