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Properties

Label 2-1456-7.4-c1-0-38
Degree $2$
Conductor $1456$
Sign $-0.268 + 0.963i$
Analytic cond. $11.6262$
Root an. cond. $3.40972$
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  + (1.13 − 1.96i)3-s + (−0.581 − 1.00i)5-s + (2.58 + 0.547i)7-s + (−1.08 − 1.87i)9-s + (−2.45 + 4.24i)11-s + 13-s − 2.64·15-s + (2.21 − 3.84i)17-s + (−3.17 − 5.49i)19-s + (4.01 − 4.47i)21-s + (−3.30 − 5.71i)23-s + (1.82 − 3.15i)25-s + 1.89·27-s + 5.27·29-s + (2.13 − 3.70i)31-s + ⋯
L(s)  = 1  + (0.655 − 1.13i)3-s + (−0.260 − 0.450i)5-s + (0.978 + 0.206i)7-s + (−0.360 − 0.624i)9-s + (−0.739 + 1.28i)11-s + 0.277·13-s − 0.682·15-s + (0.537 − 0.931i)17-s + (−0.727 − 1.25i)19-s + (0.876 − 0.975i)21-s + (−0.688 − 1.19i)23-s + (0.364 − 0.631i)25-s + 0.365·27-s + 0.979·29-s + (0.383 − 0.664i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.268 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.268 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $-0.268 + 0.963i$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ -0.268 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.089599717\)
\(L(\frac12)\) \(\approx\) \(2.089599717\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-2.58 - 0.547i)T \)
13 \( 1 - T \)
good3 \( 1 + (-1.13 + 1.96i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.581 + 1.00i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.45 - 4.24i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.21 + 3.84i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.17 + 5.49i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.30 + 5.71i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.27T + 29T^{2} \)
31 \( 1 + (-2.13 + 3.70i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.04 + 6.99i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.37T + 41T^{2} \)
43 \( 1 - 0.891T + 43T^{2} \)
47 \( 1 + (1.45 + 2.51i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.132 - 0.228i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.78 - 8.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.39 - 5.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.53 - 13.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 + (-4.59 + 7.95i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.58 + 6.20i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.79T + 83T^{2} \)
89 \( 1 + (-2.87 - 4.98i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.5T + 97T^{2} \)
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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

−8.810455371834515723680525040632, −8.522037940446262481269283647461, −7.54864004371611746411320689671, −7.25516424993418369294051798511, −6.15560933841345014876092990709, −4.84046375019317200111914854393, −4.46320720534400357212396496153, −2.62162227976436159296275203207, −2.16056544939535776130220484710, −0.807172355473882261693297176276, 1.56522869393538940469566378093, 3.14943553230145697410700670824, 3.59377558346087095353773546962, 4.56299240829525400316364049796, 5.49035629439307843631985737971, 6.36410299719600686604244600144, 7.76645361216016046395756539410, 8.228085770942407173087560868540, 8.772675384353414479651596447049, 9.941936390553422231455548215427

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