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Properties

Label 1-2415-2415.1487-r0-0-0
Degree $1$
Conductor $2415$
Sign $0.888 - 0.458i$
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  + (−0.690 − 0.723i)2-s + (−0.0475 + 0.998i)4-s + (0.755 − 0.654i)8-s + (0.723 + 0.690i)11-s + (0.909 + 0.415i)13-s + (−0.995 − 0.0950i)16-s + (−0.458 + 0.888i)17-s + (0.888 − 0.458i)19-s − i·22-s + (−0.327 − 0.945i)26-s + (0.841 − 0.540i)29-s + (0.327 − 0.945i)31-s + (0.618 + 0.786i)32-s + (0.959 − 0.281i)34-s + (0.371 − 0.928i)37-s + (−0.945 − 0.327i)38-s + ⋯
L(s)  = 1  + (−0.690 − 0.723i)2-s + (−0.0475 + 0.998i)4-s + (0.755 − 0.654i)8-s + (0.723 + 0.690i)11-s + (0.909 + 0.415i)13-s + (−0.995 − 0.0950i)16-s + (−0.458 + 0.888i)17-s + (0.888 − 0.458i)19-s − i·22-s + (−0.327 − 0.945i)26-s + (0.841 − 0.540i)29-s + (0.327 − 0.945i)31-s + (0.618 + 0.786i)32-s + (0.959 − 0.281i)34-s + (0.371 − 0.928i)37-s + (−0.945 − 0.327i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2415\)    =    \(3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $0.888 - 0.458i$
Analytic conductor: \(11.2152\)
Root analytic conductor: \(11.2152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2415} (1487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2415,\ (0:\ ),\ 0.888 - 0.458i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.283695814 - 0.3112712410i\)
\(L(\frac12)\) \(\approx\) \(1.283695814 - 0.3112712410i\)
\(L(1)\) \(\approx\) \(0.8646136242 - 0.1860220677i\)
\(L(1)\) \(\approx\) \(0.8646136242 - 0.1860220677i\)

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 \( 1 + (-0.690 - 0.723i)T \)
11 \( 1 + (0.723 + 0.690i)T \)
13 \( 1 + (0.909 + 0.415i)T \)
17 \( 1 + (-0.458 + 0.888i)T \)
19 \( 1 + (0.888 - 0.458i)T \)
29 \( 1 + (0.841 - 0.540i)T \)
31 \( 1 + (0.327 - 0.945i)T \)
37 \( 1 + (0.371 - 0.928i)T \)
41 \( 1 + (-0.142 + 0.989i)T \)
43 \( 1 + (-0.755 - 0.654i)T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (-0.814 - 0.580i)T \)
59 \( 1 + (0.995 - 0.0950i)T \)
61 \( 1 + (0.981 - 0.189i)T \)
67 \( 1 + (0.971 - 0.235i)T \)
71 \( 1 + (0.959 + 0.281i)T \)
73 \( 1 + (0.998 + 0.0475i)T \)
79 \( 1 + (0.580 + 0.814i)T \)
83 \( 1 + (0.989 - 0.142i)T \)
89 \( 1 + (-0.327 - 0.945i)T \)
97 \( 1 + (-0.989 - 0.142i)T \)
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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

−19.4152191060945746453609691957, −18.759912162650988489813654176082, −18.0003394423290892719961993892, −17.61694566022544153753601309615, −16.54173180808209277546953836936, −16.14279305189103545132656839446, −15.53872296353616632717871756172, −14.61397059865018047528365598571, −13.89506462318245849415926153197, −13.501700342757579771467526369461, −12.25001598532910406023449289424, −11.38356099095626301445933172364, −10.83625922018460152324993949911, −9.9008163175768894059201101714, −9.27274970303515064234549201595, −8.440239318554113358923855019240, −8.01266031822796175259619239945, −6.8274442539149276825666759467, −6.48878965994279840248947954304, −5.501005980294368982347421179702, −4.86642723364664586038227366034, −3.704967089001818269692371462283, −2.8145574327811801975527670036, −1.45991637085076199689785420678, −0.83431394800543921247191279786, 0.81666640528835427536453328371, 1.714746764124804244103007292482, 2.45319821301997412137454406021, 3.62413077773779351722013454585, 4.08785188984030527579731550305, 5.07707788032461726785407574195, 6.43009855121785274283597001268, 6.85184889404053616819073714229, 8.00285595786947005838503113839, 8.47023454266849952896339375427, 9.480964758872919615199850256859, 9.79033910667713034780298048609, 10.85272130799322058772475501881, 11.44344560989301413896502363060, 12.02545705014605692732364841786, 12.93899956287789286793752749234, 13.48186373362377647923690530286, 14.37520109929927938581770276240, 15.34733937582179037763745437833, 16.04025869441101368625640048352, 16.827163891956795014320194157486, 17.506103805511330401211609593, 18.05517273671466650791507020067, 18.785947768427105048771628213059, 19.59517188796981893268811838376

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