Properties

Label 2-1456-1456.307-c0-0-1
Degree $2$
Conductor $1456$
Sign $0.0985 + 0.995i$
Analytic cond. $0.726638$
Root an. cond. $0.852431$
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 + (0.707 + 0.707i)3-s − 4-s + (0.707 − 0.707i)6-s + (−0.707 − 0.707i)7-s + i·8-s + 11-s + (−0.707 − 0.707i)12-s + (−0.707 − 0.707i)13-s + (−0.707 + 0.707i)14-s + 16-s − 1.41i·19-s − 1.00i·21-s i·22-s i·23-s + (−0.707 + 0.707i)24-s + ⋯
L(s)  = 1  i·2-s + (0.707 + 0.707i)3-s − 4-s + (0.707 − 0.707i)6-s + (−0.707 − 0.707i)7-s + i·8-s + 11-s + (−0.707 − 0.707i)12-s + (−0.707 − 0.707i)13-s + (−0.707 + 0.707i)14-s + 16-s − 1.41i·19-s − 1.00i·21-s i·22-s i·23-s + (−0.707 + 0.707i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0985 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0985 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $0.0985 + 0.995i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :0),\ 0.0985 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.168033446\)
\(L(\frac12)\) \(\approx\) \(1.168033446\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
7 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
5 \( 1 - T^{2} \)
11 \( 1 - T + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + 1.41iT - T^{2} \)
23 \( 1 + iT - T^{2} \)
29 \( 1 + (-1 - i)T + iT^{2} \)
31 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
37 \( 1 + iT - T^{2} \)
41 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
43 \( 1 + (1 - i)T - iT^{2} \)
47 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 + (1 + i)T + iT^{2} \)
73 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
79 \( 1 - iT - T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
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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

−9.540880618589220159740423536692, −9.025185461229121249460919181051, −8.365639281395802407931887925178, −7.12923469345733971138753500651, −6.29868285608613215594744481905, −4.70650669453271665325662552403, −4.40881658431381794246909072262, −3.04664880330489875182169888582, −2.96678280954904086361030421509, −0.990448338193624039235069371675, 1.63655030766313258869036750528, 2.94575877607889008449941534544, 3.99005742899899529720189979456, 5.06203576954810935966537068903, 6.11880276767701033195038976551, 6.68417769564901880737495430120, 7.45800697668475361565124174411, 8.259683147057823956147059075477, 8.865756647866876781372466849419, 9.602969250482093923742990212749

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