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Properties

Label 1-2672-2672.613-r0-0-0
Degree $1$
Conductor $2672$
Sign $0.232 + 0.972i$
Analytic cond. $12.4087$
Root an. cond. $12.4087$
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  + (0.472 + 0.881i)3-s + (0.890 − 0.455i)5-s + (0.752 + 0.658i)7-s + (−0.553 + 0.832i)9-s + (0.629 + 0.776i)11-s + (0.0378 + 0.999i)13-s + (0.822 + 0.569i)15-s + (0.0567 − 0.998i)17-s + (0.739 − 0.672i)19-s + (−0.225 + 0.974i)21-s + (−0.280 + 0.959i)23-s + (0.584 − 0.811i)25-s + (−0.995 − 0.0944i)27-s + (0.959 − 0.280i)29-s + (−0.169 − 0.985i)31-s + ⋯
L(s)  = 1  + (0.472 + 0.881i)3-s + (0.890 − 0.455i)5-s + (0.752 + 0.658i)7-s + (−0.553 + 0.832i)9-s + (0.629 + 0.776i)11-s + (0.0378 + 0.999i)13-s + (0.822 + 0.569i)15-s + (0.0567 − 0.998i)17-s + (0.739 − 0.672i)19-s + (−0.225 + 0.974i)21-s + (−0.280 + 0.959i)23-s + (0.584 − 0.811i)25-s + (−0.995 − 0.0944i)27-s + (0.959 − 0.280i)29-s + (−0.169 − 0.985i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.232 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.232 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2672\)    =    \(2^{4} \cdot 167\)
Sign: $0.232 + 0.972i$
Analytic conductor: \(12.4087\)
Root analytic conductor: \(12.4087\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2672} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2672,\ (0:\ ),\ 0.232 + 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.306942157 + 1.819553100i\)
\(L(\frac12)\) \(\approx\) \(2.306942157 + 1.819553100i\)
\(L(1)\) \(\approx\) \(1.561098841 + 0.6210574645i\)
\(L(1)\) \(\approx\) \(1.561098841 + 0.6210574645i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
167 \( 1 \)
good3 \( 1 + (0.472 + 0.881i)T \)
5 \( 1 + (0.890 - 0.455i)T \)
7 \( 1 + (0.752 + 0.658i)T \)
11 \( 1 + (0.629 + 0.776i)T \)
13 \( 1 + (0.0378 + 0.999i)T \)
17 \( 1 + (0.0567 - 0.998i)T \)
19 \( 1 + (0.739 - 0.672i)T \)
23 \( 1 + (-0.280 + 0.959i)T \)
29 \( 1 + (0.959 - 0.280i)T \)
31 \( 1 + (-0.169 - 0.985i)T \)
37 \( 1 + (-0.832 + 0.553i)T \)
41 \( 1 + (0.942 - 0.334i)T \)
43 \( 1 + (0.948 - 0.316i)T \)
47 \( 1 + (0.929 + 0.369i)T \)
53 \( 1 + (0.261 - 0.965i)T \)
59 \( 1 + (-0.998 + 0.0567i)T \)
61 \( 1 + (-0.188 + 0.982i)T \)
67 \( 1 + (0.890 + 0.455i)T \)
71 \( 1 + (0.644 - 0.764i)T \)
73 \( 1 + (-0.954 - 0.298i)T \)
79 \( 1 + (0.614 - 0.788i)T \)
83 \( 1 + (0.0756 - 0.997i)T \)
89 \( 1 + (-0.822 + 0.569i)T \)
97 \( 1 + (-0.169 + 0.985i)T \)
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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

−19.119732695478604333645734765896, −18.34712990718497092644733743947, −17.807038687107204114779217055374, −17.28879605160583483386592935951, −16.587941417422096519427539023555, −15.43959498623431435081882939598, −14.52158889430654367891329208925, −14.041950293984629035263444452732, −13.83666260230483612928599431275, −12.64112069624970629661121474272, −12.33940072421432420987975920352, −11.06240769526326039958218715888, −10.65649078755557341556351208258, −9.80969940180839019819467525419, −8.782937337151825173834780797345, −8.249798890885493974057408896883, −7.49209389931731019958341871922, −6.698248768287029734491781466347, −5.99546857298296977908523538052, −5.37495929182511471653780421389, −4.04841105666723655131046796786, −3.255819413841783210683111744835, −2.4798778208898024915322750394, −1.44880786797475441163788607536, −0.98035083658046510061099052499, 1.25500368046955586071488882432, 2.15187347003478264014309041172, 2.69220408021523480039824256425, 3.9810954226428724595993769161, 4.68089462262238185275302265072, 5.22470449981185923330991485344, 6.00913149200967497220527210754, 7.10493881664611933944788604090, 7.92980745582814614194382618577, 9.07192295075636691311838635842, 9.17426133884304756558115294236, 9.8026802327623048188489188685, 10.76717874151424760308349981136, 11.716165277095578103504016656049, 12.03414545635596384978024997068, 13.3251728633146846828849479674, 14.0182815046867259786783942615, 14.311235476122183636331830398246, 15.28736306994767342734940868584, 15.8440883255135184982844182430, 16.58897877673941030772619564753, 17.46546740252157249408152483086, 17.77195907625576640973311186036, 18.79801637986544225582914282399, 19.59164397777764473542859413833

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