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Properties

Label 1-2415-2415.1433-r0-0-0
Degree $1$
Conductor $2415$
Sign $-0.547 + 0.836i$
Analytic cond. $11.2152$
Root an. cond. $11.2152$
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  + (−0.618 − 0.786i)2-s + (−0.235 + 0.971i)4-s + (0.909 − 0.415i)8-s + (−0.786 − 0.618i)11-s + (0.540 − 0.841i)13-s + (−0.888 − 0.458i)16-s + (0.690 + 0.723i)17-s + (−0.723 − 0.690i)19-s + i·22-s + (−0.995 + 0.0950i)26-s + (−0.959 − 0.281i)29-s + (0.995 + 0.0950i)31-s + (0.189 + 0.981i)32-s + (0.142 − 0.989i)34-s + (−0.945 − 0.327i)37-s + (−0.0950 + 0.995i)38-s + ⋯
L(s)  = 1  + (−0.618 − 0.786i)2-s + (−0.235 + 0.971i)4-s + (0.909 − 0.415i)8-s + (−0.786 − 0.618i)11-s + (0.540 − 0.841i)13-s + (−0.888 − 0.458i)16-s + (0.690 + 0.723i)17-s + (−0.723 − 0.690i)19-s + i·22-s + (−0.995 + 0.0950i)26-s + (−0.959 − 0.281i)29-s + (0.995 + 0.0950i)31-s + (0.189 + 0.981i)32-s + (0.142 − 0.989i)34-s + (−0.945 − 0.327i)37-s + (−0.0950 + 0.995i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2415\)    =    \(3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $-0.547 + 0.836i$
Analytic conductor: \(11.2152\)
Root analytic conductor: \(11.2152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2415} (1433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2415,\ (0:\ ),\ -0.547 + 0.836i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03200121157 - 0.05921493272i\)
\(L(\frac12)\) \(\approx\) \(-0.03200121157 - 0.05921493272i\)
\(L(1)\) \(\approx\) \(0.5749114582 - 0.2388627557i\)
\(L(1)\) \(\approx\) \(0.5749114582 - 0.2388627557i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.618 - 0.786i)T \)
11 \( 1 + (-0.786 - 0.618i)T \)
13 \( 1 + (0.540 - 0.841i)T \)
17 \( 1 + (0.690 + 0.723i)T \)
19 \( 1 + (-0.723 - 0.690i)T \)
29 \( 1 + (-0.959 - 0.281i)T \)
31 \( 1 + (0.995 + 0.0950i)T \)
37 \( 1 + (-0.945 - 0.327i)T \)
41 \( 1 + (-0.654 + 0.755i)T \)
43 \( 1 + (-0.909 - 0.415i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (-0.998 + 0.0475i)T \)
59 \( 1 + (0.888 - 0.458i)T \)
61 \( 1 + (0.580 - 0.814i)T \)
67 \( 1 + (-0.371 + 0.928i)T \)
71 \( 1 + (0.142 + 0.989i)T \)
73 \( 1 + (-0.971 - 0.235i)T \)
79 \( 1 + (0.0475 - 0.998i)T \)
83 \( 1 + (-0.755 + 0.654i)T \)
89 \( 1 + (-0.995 + 0.0950i)T \)
97 \( 1 + (0.755 + 0.654i)T \)
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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

−19.825611646646088985327802407680, −18.95681505645197027432274757740, −18.567357201113055414958342876603, −17.90639248420095444698124105313, −17.01814771864973026274542601846, −16.504032203271161834905532980897, −15.78832990754713290454331879167, −15.12685785389914342016562630441, −14.405817510678254475708916218701, −13.71031534129219200848075571881, −12.97416975805613235931825680906, −11.95365843945415721866131684171, −11.18313334869093859361717389910, −10.20255262603899738135290022301, −9.87139914543124811756213551220, −8.85542377521859853021739923616, −8.27179434896687771506719827052, −7.43870422236991491526491114822, −6.81182547541466299269476739128, −5.993713940614402251911992018, −5.16538440435455732445376580524, −4.49139862886140488296591446448, −3.40397247214163129238659095797, −2.1160177074055503186050981573, −1.41743929334705761810783458790, 0.028854208876460906353583372419, 1.13445985807939621677333467611, 2.09593268960132435369863443829, 3.07300564799741925240829047528, 3.57313954128512119605359873730, 4.66039533211765533644968508971, 5.549776341167186671916377051440, 6.49432702629241833460908873845, 7.509191484990061145104786481633, 8.34902086109212907927105096487, 8.55731626306266899515219282136, 9.79705739699975034862334857457, 10.30292807484531156571429030300, 11.02581286069370595471378043261, 11.589291282731897374225267764142, 12.66435561532084520127593780272, 13.05947361455617174826177538409, 13.73572070774365297897452465945, 14.81991759848122547455749468392, 15.66917081603966287141575376603, 16.28617304940711178094465364276, 17.22564778352832668874899064378, 17.60651944494861916393410587631, 18.57905918900736549975861186693, 18.98190248536670493008876323707

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