Properties

Label 1-77-77.19-r0-0-0
Degree $1$
Conductor $77$
Sign $0.780 + 0.625i$
Analytic cond. $0.357586$
Root an. cond. $0.357586$
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  + (0.978 − 0.207i)2-s + (0.104 + 0.994i)3-s + (0.913 − 0.406i)4-s + (−0.669 + 0.743i)5-s + (0.309 + 0.951i)6-s + (0.809 − 0.587i)8-s + (−0.978 + 0.207i)9-s + (−0.5 + 0.866i)10-s + (0.5 + 0.866i)12-s + (0.309 − 0.951i)13-s + (−0.809 − 0.587i)15-s + (0.669 − 0.743i)16-s + (−0.978 − 0.207i)17-s + (−0.913 + 0.406i)18-s + (0.913 + 0.406i)19-s + (−0.309 + 0.951i)20-s + ⋯
L(s)  = 1  + (0.978 − 0.207i)2-s + (0.104 + 0.994i)3-s + (0.913 − 0.406i)4-s + (−0.669 + 0.743i)5-s + (0.309 + 0.951i)6-s + (0.809 − 0.587i)8-s + (−0.978 + 0.207i)9-s + (−0.5 + 0.866i)10-s + (0.5 + 0.866i)12-s + (0.309 − 0.951i)13-s + (−0.809 − 0.587i)15-s + (0.669 − 0.743i)16-s + (−0.978 − 0.207i)17-s + (−0.913 + 0.406i)18-s + (0.913 + 0.406i)19-s + (−0.309 + 0.951i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.780 + 0.625i$
Analytic conductor: \(0.357586\)
Root analytic conductor: \(0.357586\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 77,\ (0:\ ),\ 0.780 + 0.625i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.439034909 + 0.5056594432i\)
\(L(\frac12)\) \(\approx\) \(1.439034909 + 0.5056594432i\)
\(L(1)\) \(\approx\) \(1.537454266 + 0.3434537110i\)
\(L(1)\) \(\approx\) \(1.537454266 + 0.3434537110i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.978 - 0.207i)T \)
3 \( 1 + (0.104 + 0.994i)T \)
5 \( 1 + (-0.669 + 0.743i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (-0.978 - 0.207i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (-0.104 + 0.994i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.913 - 0.406i)T \)
53 \( 1 + (0.669 + 0.743i)T \)
59 \( 1 + (-0.913 + 0.406i)T \)
61 \( 1 + (0.669 - 0.743i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.913 - 0.406i)T \)
79 \( 1 + (0.978 - 0.207i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.309 + 0.951i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−31.09034868557911669644776785408, −30.5061763210939678161150754400, −29.09503392005370624473925175710, −28.45155651379382212365223781043, −26.57488491231435590694060378930, −25.36668176157420657877981720651, −24.29468175163218112550122773091, −23.80575484497749629398142381312, −22.80087467423483546345355280245, −21.40442273589410092475056017837, −20.097401747441990912577384444562, −19.46700593025469271624612334803, −17.799562061281194033077897414789, −16.50437279048455829963274167553, −15.47439236746195381532791904633, −14.01044335487236869196670567991, −13.185686941585511132578034824425, −12.04977899383917054584891720288, −11.340084783254581605318337736231, −8.82517761664321025769127806442, −7.617862121809934456506989449226, −6.53513849373176582042958147526, −5.063526359236703429196612445133, −3.596905962077916445277963492252, −1.80112192893565498596555615091, 2.79197633050398092454864021373, 3.782828880218074418663497166078, 5.047824120874267938369599439143, 6.50523692340066600912082450782, 8.11236172854510801151714257314, 10.06356458110452411402937494865, 10.95425357454686060644987999383, 11.95360380606866358921819419468, 13.59668011146540732024113416282, 14.75704084924762042246074063564, 15.49178802325851837450168786113, 16.41345726148107198029252059755, 18.3022809399873708937103679466, 19.911760680456833906322773844369, 20.42332934561924892683370635956, 21.92319077813511280002018825364, 22.477376631078758291122520997297, 23.4102334623861643576376543155, 24.839668300483075640763661816210, 26.06036776503719882975073223108, 27.11847963641426456424247104917, 28.189526049467945799792484282345, 29.398995757305853721640132438, 30.71041975809324079114918249195, 31.2866493468988398254269226582

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