Properties

Label 1-2415-2415.584-r0-0-0
Degree $1$
Conductor $2415$
Sign $0.624 - 0.780i$
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  + (0.0475 + 0.998i)2-s + (−0.995 + 0.0950i)4-s + (−0.142 − 0.989i)8-s + (−0.0475 + 0.998i)11-s + (−0.654 + 0.755i)13-s + (0.981 − 0.189i)16-s + (−0.580 + 0.814i)17-s + (−0.580 − 0.814i)19-s − 22-s + (−0.786 − 0.618i)26-s + (−0.415 − 0.909i)29-s + (0.786 − 0.618i)31-s + (0.235 + 0.971i)32-s + (−0.841 − 0.540i)34-s + (−0.723 + 0.690i)37-s + (0.786 − 0.618i)38-s + ⋯
L(s)  = 1  + (0.0475 + 0.998i)2-s + (−0.995 + 0.0950i)4-s + (−0.142 − 0.989i)8-s + (−0.0475 + 0.998i)11-s + (−0.654 + 0.755i)13-s + (0.981 − 0.189i)16-s + (−0.580 + 0.814i)17-s + (−0.580 − 0.814i)19-s − 22-s + (−0.786 − 0.618i)26-s + (−0.415 − 0.909i)29-s + (0.786 − 0.618i)31-s + (0.235 + 0.971i)32-s + (−0.841 − 0.540i)34-s + (−0.723 + 0.690i)37-s + (0.786 − 0.618i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2415\)    =    \(3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $0.624 - 0.780i$
Analytic conductor: \(11.2152\)
Root analytic conductor: \(11.2152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2415} (584, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2415,\ (0:\ ),\ 0.624 - 0.780i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2695900829 - 0.1295488996i\)
\(L(\frac12)\) \(\approx\) \(0.2695900829 - 0.1295488996i\)
\(L(1)\) \(\approx\) \(0.6544098621 + 0.3832360671i\)
\(L(1)\) \(\approx\) \(0.6544098621 + 0.3832360671i\)

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 \( 1 + (0.0475 + 0.998i)T \)
11 \( 1 + (-0.0475 + 0.998i)T \)
13 \( 1 + (-0.654 + 0.755i)T \)
17 \( 1 + (-0.580 + 0.814i)T \)
19 \( 1 + (-0.580 - 0.814i)T \)
29 \( 1 + (-0.415 - 0.909i)T \)
31 \( 1 + (0.786 - 0.618i)T \)
37 \( 1 + (-0.723 + 0.690i)T \)
41 \( 1 + (-0.959 + 0.281i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.327 + 0.945i)T \)
59 \( 1 + (0.981 + 0.189i)T \)
61 \( 1 + (-0.928 - 0.371i)T \)
67 \( 1 + (0.888 + 0.458i)T \)
71 \( 1 + (-0.841 + 0.540i)T \)
73 \( 1 + (-0.995 + 0.0950i)T \)
79 \( 1 + (-0.327 - 0.945i)T \)
83 \( 1 + (0.959 + 0.281i)T \)
89 \( 1 + (-0.786 - 0.618i)T \)
97 \( 1 + (-0.959 + 0.281i)T \)
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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

−19.466294606768407277675597008176, −19.211947069563206153464084658242, −18.24409274199883908725508900841, −17.75129236824510102895793906226, −16.90899180224426581288091625741, −16.15190617921901322251122870658, −15.21346062041488517939879195357, −14.341429303619388506822188459903, −13.85163988149793588785920269713, −12.97125281993154517922679047371, −12.427342085691737733679802224904, −11.63102087552533088289940926083, −10.89767687732802484820295318887, −10.32967229438081739887587850193, −9.5342555057921189263396659847, −8.70194446610438707804997847217, −8.14793425222217625425201183412, −7.14212134946663742397125046674, −6.026176181700039132492578581298, −5.26390461469798136166262860059, −4.55370284139189599706440170879, −3.50896616580485742385626386386, −2.94065219795300881806118631947, −2.03716491167953787253939074265, −0.99272457149206374870664300735, 0.10758879548882208721301459573, 1.67418573654929740109837379440, 2.58893670171307046224086136077, 3.963969631060848948076633264832, 4.454584348217223849450171743981, 5.19631601045093368339428215022, 6.21595235922755124692597803283, 6.853513220306913574650497777268, 7.43999858861358310642578910653, 8.38360302505647644913565191516, 9.01195996934730821717873816806, 9.826723167532715988810035441538, 10.4271188044971029342943879349, 11.64895285790583631760464036456, 12.3207913907446156170155071303, 13.187618753896132615329821452693, 13.67919741642718723823567370312, 14.646019208682944672961774968023, 15.24778248090284037343757556024, 15.59312963929418872533370038840, 16.76130920773261638738138494167, 17.2487737785381253149467418243, 17.6446599447554205553672237817, 18.75062244162137281604407476397, 19.15757349144196555552372868818

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