Properties

Label 1-2015-2015.1332-r1-0-0
Degree $1$
Conductor $2015$
Sign $0.870 + 0.492i$
Analytic cond. $216.541$
Root an. cond. $216.541$
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  + (−0.5 − 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.5 + 0.866i)4-s + (−0.866 − 0.5i)6-s + (0.5 − 0.866i)7-s + 8-s + (0.5 − 0.866i)9-s + (−0.866 + 0.5i)11-s + i·12-s − 14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s − 18-s + (−0.866 − 0.5i)19-s i·21-s + (0.866 + 0.5i)22-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.5 + 0.866i)4-s + (−0.866 − 0.5i)6-s + (0.5 − 0.866i)7-s + 8-s + (0.5 − 0.866i)9-s + (−0.866 + 0.5i)11-s + i·12-s − 14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s − 18-s + (−0.866 − 0.5i)19-s i·21-s + (0.866 + 0.5i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.870 + 0.492i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (1332, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ 0.870 + 0.492i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3440239996 + 0.09059023470i\)
\(L(\frac12)\) \(\approx\) \(0.3440239996 + 0.09059023470i\)
\(L(1)\) \(\approx\) \(0.6953233973 - 0.5284099893i\)
\(L(1)\) \(\approx\) \(0.6953233973 - 0.5284099893i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 - T \)
53 \( 1 + iT \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 - T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−19.39631011362636565796751453593, −18.9762518238169172542641251730, −18.2381842747892701416465116999, −17.599492886007639436819051435009, −16.51208956247995913754501630024, −16.02990825780905347390831338556, −15.223685570965941499684457942466, −14.87107055858877526522598149076, −14.09187861145893366722676614694, −13.336406504237080555381475139241, −12.57963678048433192327595853122, −11.24139328735014219364236738271, −10.51173016040862354773804765198, −9.92610328319718046943166406831, −8.834092077775252077453262124793, −8.498407528560494796806971874610, −7.99457183877317973810452050839, −6.968674267502631435674042572626, −6.00467728589785380630186048045, −5.222934690806571207999880467957, −4.51986280899641920488363625209, −3.53793529749428205003071215720, −2.28254022764623420457173205920, −1.746382456033506430075723156148, −0.07341061621598118389809384062, 0.80556206028019645326556119243, 1.983927417708072351708823879036, 2.305676040725431798862365012680, 3.443914086568141435946814333132, 4.20788984873910814932831498859, 4.908226329179404412186803339634, 6.48441610295597822105491408121, 7.42356293758535420315758715186, 7.81027078982070465317618709050, 8.5979093092747672018752002646, 9.40847367240124782502642768190, 10.11341906988589178791049466501, 10.869183827562024296957135521039, 11.62112304632410466332096398371, 12.52911569650530346641786687975, 13.29366277792051420378495452400, 13.5956008823810883048165228951, 14.50521356862344960106911816529, 15.39830308785140406637326276663, 16.22624845871530519799799634764, 17.40195730823077872374324941835, 17.72865999369219253164569217902, 18.418941118390652587932339312406, 19.234498293911575864518397324993, 19.8929614080920494559953344202

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