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Properties

Label 1-2415-2415.1139-r0-0-0
Degree $1$
Conductor $2415$
Sign $0.986 - 0.166i$
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  + (−0.995 + 0.0950i)2-s + (0.981 − 0.189i)4-s + (−0.959 + 0.281i)8-s + (0.995 + 0.0950i)11-s + (−0.142 − 0.989i)13-s + (0.928 − 0.371i)16-s + (0.327 + 0.945i)17-s + (0.327 − 0.945i)19-s − 22-s + (0.235 + 0.971i)26-s + (0.654 − 0.755i)29-s + (−0.235 + 0.971i)31-s + (−0.888 + 0.458i)32-s + (−0.415 − 0.909i)34-s + (−0.0475 + 0.998i)37-s + (−0.235 + 0.971i)38-s + ⋯
L(s)  = 1  + (−0.995 + 0.0950i)2-s + (0.981 − 0.189i)4-s + (−0.959 + 0.281i)8-s + (0.995 + 0.0950i)11-s + (−0.142 − 0.989i)13-s + (0.928 − 0.371i)16-s + (0.327 + 0.945i)17-s + (0.327 − 0.945i)19-s − 22-s + (0.235 + 0.971i)26-s + (0.654 − 0.755i)29-s + (−0.235 + 0.971i)31-s + (−0.888 + 0.458i)32-s + (−0.415 − 0.909i)34-s + (−0.0475 + 0.998i)37-s + (−0.235 + 0.971i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2415\)    =    \(3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $0.986 - 0.166i$
Analytic conductor: \(11.2152\)
Root analytic conductor: \(11.2152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2415} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2415,\ (0:\ ),\ 0.986 - 0.166i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.158369357 - 0.09708610720i\)
\(L(\frac12)\) \(\approx\) \(1.158369357 - 0.09708610720i\)
\(L(1)\) \(\approx\) \(0.7980798781 + 0.008956424583i\)
\(L(1)\) \(\approx\) \(0.7980798781 + 0.008956424583i\)

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 \( 1 + (-0.995 + 0.0950i)T \)
11 \( 1 + (0.995 + 0.0950i)T \)
13 \( 1 + (-0.142 - 0.989i)T \)
17 \( 1 + (0.327 + 0.945i)T \)
19 \( 1 + (0.327 - 0.945i)T \)
29 \( 1 + (0.654 - 0.755i)T \)
31 \( 1 + (-0.235 + 0.971i)T \)
37 \( 1 + (-0.0475 + 0.998i)T \)
41 \( 1 + (0.841 - 0.540i)T \)
43 \( 1 + (0.959 + 0.281i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.786 - 0.618i)T \)
59 \( 1 + (0.928 + 0.371i)T \)
61 \( 1 + (-0.723 - 0.690i)T \)
67 \( 1 + (-0.580 - 0.814i)T \)
71 \( 1 + (-0.415 + 0.909i)T \)
73 \( 1 + (0.981 - 0.189i)T \)
79 \( 1 + (-0.786 + 0.618i)T \)
83 \( 1 + (-0.841 - 0.540i)T \)
89 \( 1 + (0.235 + 0.971i)T \)
97 \( 1 + (0.841 - 0.540i)T \)
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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

−19.42728477904563562907842209775, −18.83052122698951558203871954588, −18.19617654114416279527435701371, −17.45054286455505527383173382085, −16.58122414332051030996893008654, −16.36840594644160551937391215002, −15.47081005649478609300549391834, −14.37762886754695660041437180359, −14.17215807326662216105229755156, −12.839018656226181640830278915891, −11.971047248445768635394857164232, −11.63461330761555084197097670863, −10.771758993236153740424205205513, −9.91224876020862540082121038038, −9.23651557481918639566731409850, −8.805391579229309057135095526455, −7.68639234696419510508331682370, −7.18536675511735951890853776056, −6.33717347478535405630698395654, −5.62427784136155574521242177385, −4.3535490408025884433245646577, −3.54745901094017466784255995306, −2.55839277642944137304996540461, −1.673677718464816493639376971674, −0.82614181603331389951002874696, 0.75371158513808251810548384323, 1.52819192135943140033454004145, 2.62861618641361735353416668196, 3.368499832524781794049573530331, 4.47981051857679555683769275160, 5.581693884680470862565178697415, 6.28466136714195738977247205540, 7.033112734814039783770672681856, 7.83641122847414662391372630053, 8.51320245886317412807385312283, 9.26714726821327792690786863414, 9.95156161651297376890118754199, 10.69544251308150536746159022409, 11.34959798330747912792776369215, 12.21762952547787314734245415609, 12.766759311308624740531538544037, 13.95348025165942340855179467732, 14.6675544615236998935114991035, 15.416736272261839508843967477607, 15.94163938424629183184825028095, 16.89817775978311256347726014607, 17.50494221254319746654807211597, 17.818041366602673353950884754775, 18.90262480230447875530766291121, 19.50140655358982831936786690746

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