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Properties

Label 1-775-775.246-r1-0-0
Degree $1$
Conductor $775$
Sign $-0.226 + 0.974i$
Analytic cond. $83.2853$
Root an. cond. $83.2853$
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  + (0.309 − 0.951i)2-s − 3-s + (−0.809 − 0.587i)4-s + (−0.309 + 0.951i)6-s + (−0.809 + 0.587i)7-s + (−0.809 + 0.587i)8-s + 9-s − 11-s + (0.809 + 0.587i)12-s + (−0.309 − 0.951i)13-s + (0.309 + 0.951i)14-s + (0.309 + 0.951i)16-s + (0.809 − 0.587i)17-s + (0.309 − 0.951i)18-s + 19-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s − 3-s + (−0.809 − 0.587i)4-s + (−0.309 + 0.951i)6-s + (−0.809 + 0.587i)7-s + (−0.809 + 0.587i)8-s + 9-s − 11-s + (0.809 + 0.587i)12-s + (−0.309 − 0.951i)13-s + (0.309 + 0.951i)14-s + (0.309 + 0.951i)16-s + (0.809 − 0.587i)17-s + (0.309 − 0.951i)18-s + 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.226 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.226 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $-0.226 + 0.974i$
Analytic conductor: \(83.2853\)
Root analytic conductor: \(83.2853\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (246, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 775,\ (1:\ ),\ -0.226 + 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2071042700 - 0.2607718061i\)
\(L(\frac12)\) \(\approx\) \(-0.2071042700 - 0.2607718061i\)
\(L(1)\) \(\approx\) \(0.5324284185 - 0.3943624148i\)
\(L(1)\) \(\approx\) \(0.5324284185 - 0.3943624148i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 - T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 - T \)
13 \( 1 + (-0.309 - 0.951i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.309 - 0.951i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (-0.309 - 0.951i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.309 + 0.951i)T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−23.04100420835725817657851585281, −21.91161574991352889981063247828, −21.67989763362594606239742518742, −20.51015828467729083083098035904, −19.18154504197135868710963190384, −18.50994211363911077723024103058, −17.64109121799046109935899670214, −16.92505983942294273000221505386, −16.1319331586725633727648197675, −15.86417628898931783265615768382, −14.65513915547221247455297449627, −13.66493955452103378619494001132, −13.00883136544175187075052497554, −12.22799479336610288356163216924, −11.33266194226004658579567006130, −9.97029486184166303229432353023, −9.72201776684860869678546876670, −8.1819493054608617321004169060, −7.30596434551884945489583585539, −6.670472772317876235718189394465, −5.75869754784795823392149255187, −5.0237150072933230700166121916, −4.07290142170851199208923279904, −3.09141538005267172583623262532, −1.1654679699999107397798336099, 0.11367855439703157365544869420, 0.85575635088386954003412732684, 2.42496178129698172620826188473, 3.14840677121026700974771669517, 4.400414848921975573204367735449, 5.533317514642531246233554683055, 5.66556808739392379725033439376, 7.074808969374887749277139238799, 8.230187327252958491758192598261, 9.53313729044213981773743107363, 10.14278182845173426676436006504, 10.745959908704231480620985924291, 11.95786349461153879844634110678, 12.28863363357674241028210598892, 13.0821491096574722004721201913, 13.90266894599140250888572756762, 15.2272113789194636253766284326, 15.81727395035855460980175916823, 16.769148145774923873061874334776, 17.91343316209692243610921022541, 18.396061482881823050628269854893, 19.02260762287039276618170935676, 20.12537921072741226856910978473, 20.85400972985281265101124549586, 21.73078817156340124960940770425

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