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Properties

Label 1-912-912.221-r0-0-0
Degree $1$
Conductor $912$
Sign $0.973 - 0.226i$
Analytic cond. $4.23531$
Root an. cond. $4.23531$
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  + (−0.866 − 0.5i)5-s − 7-s − i·11-s + (−0.866 + 0.5i)13-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)23-s + (0.5 + 0.866i)25-s + (0.866 − 0.5i)29-s − 31-s + (0.866 + 0.5i)35-s − i·37-s + (0.5 − 0.866i)41-s + (0.866 + 0.5i)43-s + (0.5 + 0.866i)47-s + 49-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)5-s − 7-s − i·11-s + (−0.866 + 0.5i)13-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)23-s + (0.5 + 0.866i)25-s + (0.866 − 0.5i)29-s − 31-s + (0.866 + 0.5i)35-s − i·37-s + (0.5 − 0.866i)41-s + (0.866 + 0.5i)43-s + (0.5 + 0.866i)47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.973 - 0.226i$
Analytic conductor: \(4.23531\)
Root analytic conductor: \(4.23531\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 912,\ (0:\ ),\ 0.973 - 0.226i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8569018015 - 0.09845599564i\)
\(L(\frac12)\) \(\approx\) \(0.8569018015 - 0.09845599564i\)
\(L(1)\) \(\approx\) \(0.7800679379 - 0.03705828771i\)
\(L(1)\) \(\approx\) \(0.7800679379 - 0.03705828771i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 - T \)
11 \( 1 - iT \)
13 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 - T \)
37 \( 1 - iT \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + (0.866 - 0.5i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−22.00884072959517005944270429484, −21.37464166732690795067918207699, −19.97525442166011720559580008179, −19.56246325774371171105322504197, −18.996188557763887976113894449945, −18.10526022839453530199098860724, −17.08015838498989843542183330539, −16.20349322019607295211098784819, −15.70898391947741252085527279788, −14.746347801155697751172194655218, −14.0655086472582431314165971989, −12.903686015257974625682998796, −12.35364438866806503584237486222, −11.39339621350901096130191340177, −10.567580155461277777600757026967, −9.8318262001795662194800394083, −8.75971342315880426704521137143, −7.863890474062109296241522220224, −7.119130549450705490736277879881, −6.161807624165689922558406248, −5.33605355138730391881429687932, −3.87733811261823653175831079513, −3.39842411387194488678912181051, −2.42378921625099929360273935959, −0.68432437821069686440693692307, 0.63481281095364721036726756977, 2.198482500259202934151453604, 3.19523425561583516318058875010, 4.29461408037224297130873672414, 4.877601139253991892286181937348, 6.154373866795922510160985968, 7.170234698735157440139602101003, 7.67315052564891349473168469814, 8.9127557630661030648835080540, 9.58639570473701393989113356886, 10.359553917043611496719403362626, 11.599277118972353595827531268162, 12.34662125437604727989196758552, 12.692058370090467027522187963270, 13.90834651210447016239969121131, 14.76045832766472400688670121672, 15.66024801248951610504655158394, 16.26202546288409635924308700700, 16.93753631939667688746068797640, 17.93171948463588232048865353388, 18.98885762292270237355902561520, 19.44258452399404446525518038362, 20.31763383698434788646564271952, 20.81555687985699265440855448807, 22.17530006461936976279443687668

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