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Properties

Label 1-2415-2415.1118-r0-0-0
Degree $1$
Conductor $2415$
Sign $0.992 - 0.126i$
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  + (0.458 − 0.888i)2-s + (−0.580 − 0.814i)4-s + (−0.989 + 0.142i)8-s + (−0.888 + 0.458i)11-s + (0.755 + 0.654i)13-s + (−0.327 + 0.945i)16-s + (0.0950 − 0.995i)17-s + (0.995 − 0.0950i)19-s + i·22-s + (0.928 − 0.371i)26-s + (0.415 + 0.909i)29-s + (−0.928 − 0.371i)31-s + (0.690 + 0.723i)32-s + (−0.841 − 0.540i)34-s + (−0.971 + 0.235i)37-s + (0.371 − 0.928i)38-s + ⋯
L(s)  = 1  + (0.458 − 0.888i)2-s + (−0.580 − 0.814i)4-s + (−0.989 + 0.142i)8-s + (−0.888 + 0.458i)11-s + (0.755 + 0.654i)13-s + (−0.327 + 0.945i)16-s + (0.0950 − 0.995i)17-s + (0.995 − 0.0950i)19-s + i·22-s + (0.928 − 0.371i)26-s + (0.415 + 0.909i)29-s + (−0.928 − 0.371i)31-s + (0.690 + 0.723i)32-s + (−0.841 − 0.540i)34-s + (−0.971 + 0.235i)37-s + (0.371 − 0.928i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2415\)    =    \(3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $0.992 - 0.126i$
Analytic conductor: \(11.2152\)
Root analytic conductor: \(11.2152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2415} (1118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2415,\ (0:\ ),\ 0.992 - 0.126i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.477600579 - 0.09361240835i\)
\(L(\frac12)\) \(\approx\) \(1.477600579 - 0.09361240835i\)
\(L(1)\) \(\approx\) \(1.064698019 - 0.4203423908i\)
\(L(1)\) \(\approx\) \(1.064698019 - 0.4203423908i\)

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 \( 1 + (0.458 - 0.888i)T \)
11 \( 1 + (-0.888 + 0.458i)T \)
13 \( 1 + (0.755 + 0.654i)T \)
17 \( 1 + (0.0950 - 0.995i)T \)
19 \( 1 + (0.995 - 0.0950i)T \)
29 \( 1 + (0.415 + 0.909i)T \)
31 \( 1 + (-0.928 - 0.371i)T \)
37 \( 1 + (-0.971 + 0.235i)T \)
41 \( 1 + (-0.959 + 0.281i)T \)
43 \( 1 + (0.989 + 0.142i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (0.189 + 0.981i)T \)
59 \( 1 + (0.327 + 0.945i)T \)
61 \( 1 + (-0.786 + 0.618i)T \)
67 \( 1 + (-0.998 + 0.0475i)T \)
71 \( 1 + (-0.841 + 0.540i)T \)
73 \( 1 + (0.814 - 0.580i)T \)
79 \( 1 + (0.981 + 0.189i)T \)
83 \( 1 + (-0.281 + 0.959i)T \)
89 \( 1 + (0.928 - 0.371i)T \)
97 \( 1 + (0.281 + 0.959i)T \)
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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

−19.44161639992686269007944223987, −18.59904344079800782678902355530, −17.98835434441903383766860006400, −17.38582985993938745421685726626, −16.50470025521681174814605063573, −15.8564870422899898616853772893, −15.409769687198242684111966280986, −14.56838240165882148627651839778, −13.772908953268301320794032478696, −13.231491758008355866170789921848, −12.5762526740107286505810824701, −11.75038801513068460794379953772, −10.79518983551226377140096953424, −10.07432116200424504890647356883, −9.03338362472048217597725855802, −8.276699674397328993097599317732, −7.81466073594394512899616990535, −6.91922870750442678531330504296, −5.99202135905274436046026964625, −5.50698138961699595964923555353, −4.73148845068251487777010985950, −3.54400320383728867133514995976, −3.251805611552465922551819915151, −1.91811902117650304666548400457, −0.46714408538860255487531066336, 1.004616317390524268284734799469, 1.8689701249553070191855871355, 2.83366809173787770830253005084, 3.45756432342868872276956704237, 4.49983629060590552639758396722, 5.11768406290369068795644465267, 5.85652691884620444074565869361, 6.88914684206411925207345103619, 7.69047952604755472600742318517, 8.8393690246598612140103590159, 9.34415013048279032160409926579, 10.224782677939608810781590767694, 10.83785592653573934565834627562, 11.639729726750909895169550398317, 12.189219179124647774475826187759, 13.06647536090003742440726112920, 13.67973260824694636275551454027, 14.229496364644648778650370487864, 15.140150768180673946757484373470, 15.87110954375746624066778512475, 16.50763584812953430811951010303, 17.78961341069414378475606271681, 18.25360291415806965122734005654, 18.74503295999962005958987730610, 19.69018970865187264323883674363

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