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Properties

Label 1-259-259.228-r1-0-0
Degree $1$
Conductor $259$
Sign $-0.993 - 0.115i$
Analytic cond. $27.8334$
Root an. cond. $27.8334$
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  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s + (−0.866 − 0.5i)5-s + i·6-s + i·8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + i·13-s − i·15-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.866 + 0.5i)18-s + (−0.866 − 0.5i)19-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s + (−0.866 − 0.5i)5-s + i·6-s + i·8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + i·13-s − i·15-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.866 + 0.5i)18-s + (−0.866 − 0.5i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.993 - 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.993 - 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(259\)    =    \(7 \cdot 37\)
Sign: $-0.993 - 0.115i$
Analytic conductor: \(27.8334\)
Root analytic conductor: \(27.8334\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{259} (228, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 259,\ (1:\ ),\ -0.993 - 0.115i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1416750439 + 2.443102982i\)
\(L(\frac12)\) \(\approx\) \(-0.1416750439 + 2.443102982i\)
\(L(1)\) \(\approx\) \(1.136144953 + 1.153395389i\)
\(L(1)\) \(\approx\) \(1.136144953 + 1.153395389i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + iT \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 - iT \)
31 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 - T \)
43 \( 1 + iT \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.866 - 0.5i)T \)
61 \( 1 + (0.866 + 0.5i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (-0.866 - 0.5i)T \)
83 \( 1 + T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + iT \)
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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

−25.06826499042844200092242911038, −23.90685820746362783294667663345, −23.5680694796566619616364987789, −22.5433137765094602987467046546, −21.64747057541860864836494932992, −20.39120189225109408700373098662, −19.73794034228402561612849368489, −18.97932900331628816113035336642, −18.303493335267308900091251660114, −16.71298881035675485295494602582, −15.405446996434092532212602655526, −14.66430761392558957013351096418, −13.935739743598205543609357153706, −12.80001341843554285025085829270, −12.122696433799218889180520011822, −11.19159321199826131971325040570, −10.17716524135329586302264270431, −8.56239025571343331937050730212, −7.61317057703457106695973571999, −6.50930006738950626755872929907, −5.58009630859033266594501968933, −3.71294229807211345183707271334, −3.31541143254122423343618727207, −1.885506627951272129590556838226, −0.507993528145774757520834394540, 2.16342386429120444209669581470, 3.61277962485321863743629542675, 4.335487750749499887319115571556, 5.080511317485468232178567390946, 6.64168238423615275938691996822, 7.76488280315421856432853900528, 8.648117514052133716282258814203, 9.731778161688205903647705789468, 11.24876413880757268638599230224, 12.00253425146940670353013070963, 13.0533212444212289476971781570, 14.3309385378250535350706334777, 14.82017077220680875624811812496, 15.887504460138176183810386397681, 16.424101990130570959179291601968, 17.33910573464260868950729743378, 19.115834911998857719365067847330, 20.05980919413928040348719720363, 20.75339631712222006998202586213, 21.596048073092446944228016657679, 22.57374496353307159955624324198, 23.35091047061278185055812245482, 24.25242061635604291100358727856, 25.27368201877838451959691472608, 25.98124430841122404999196944437

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