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Properties

Label 2-1456-13.9-c1-0-10
Degree $2$
Conductor $1456$
Sign $-0.0128 - 0.999i$
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  − 5-s + (−0.5 − 0.866i)7-s + (1.5 + 2.59i)9-s + (−1 + 1.73i)11-s + (3.5 + 0.866i)13-s + (1.5 + 2.59i)17-s + (−3 − 5.19i)19-s + (−2 + 3.46i)23-s − 4·25-s + (3.5 − 6.06i)29-s − 4·31-s + (0.5 + 0.866i)35-s + (−4.5 + 7.79i)37-s + (−4.5 + 7.79i)41-s + (5 + 8.66i)43-s + ⋯
L(s)  = 1  − 0.447·5-s + (−0.188 − 0.327i)7-s + (0.5 + 0.866i)9-s + (−0.301 + 0.522i)11-s + (0.970 + 0.240i)13-s + (0.363 + 0.630i)17-s + (−0.688 − 1.19i)19-s + (−0.417 + 0.722i)23-s − 0.800·25-s + (0.649 − 1.12i)29-s − 0.718·31-s + (0.0845 + 0.146i)35-s + (−0.739 + 1.28i)37-s + (−0.702 + 1.21i)41-s + (0.762 + 1.32i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $-0.0128 - 0.999i$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ -0.0128 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.190326568\)
\(L(\frac12)\) \(\approx\) \(1.190326568\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-3.5 - 0.866i)T \)
good3 \( 1 + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + T + 5T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.5 + 6.06i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (4.5 - 7.79i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2T + 47T^{2} \)
53 \( 1 - 9T + 53T^{2} \)
59 \( 1 + (-7 - 12.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5 - 8.66i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 + 2T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.957092576590929367851818577979, −8.784912819514252520236630864563, −8.063469087569654328525366741188, −7.38559778192496063108922249683, −6.55317747396699365239383293319, −5.59314324856627679174299047848, −4.47295037722606286586029578956, −3.94016288269780118022406740548, −2.61643743439669012360120528646, −1.40151883700983408992146203129, 0.49870685824528081504784984928, 2.01801590895761174109505791876, 3.55648629853055582686055585539, 3.82107033005566794003713054272, 5.28790517537526930567765011410, 6.02302168340223156310534432405, 6.85669009648069150113453610028, 7.74811084883029840071587525329, 8.629262595615513338256919224167, 9.114057566145506150334802496250

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