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Properties

Label 1-3328-3328.1051-r0-0-0
Degree $1$
Conductor $3328$
Sign $-0.224 + 0.974i$
Analytic cond. $15.4551$
Root an. cond. $15.4551$
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  + (−0.0327 + 0.999i)3-s + (0.773 − 0.634i)5-s + (−0.442 + 0.896i)7-s + (−0.997 − 0.0654i)9-s + (0.973 + 0.227i)11-s + (0.608 + 0.793i)15-s + (0.793 + 0.608i)17-s + (0.582 + 0.812i)19-s + (−0.881 − 0.471i)21-s + (−0.321 + 0.946i)23-s + (0.195 − 0.980i)25-s + (0.0980 − 0.995i)27-s + (0.683 + 0.729i)29-s + (0.707 − 0.707i)31-s + (−0.258 + 0.965i)33-s + ⋯
L(s)  = 1  + (−0.0327 + 0.999i)3-s + (0.773 − 0.634i)5-s + (−0.442 + 0.896i)7-s + (−0.997 − 0.0654i)9-s + (0.973 + 0.227i)11-s + (0.608 + 0.793i)15-s + (0.793 + 0.608i)17-s + (0.582 + 0.812i)19-s + (−0.881 − 0.471i)21-s + (−0.321 + 0.946i)23-s + (0.195 − 0.980i)25-s + (0.0980 − 0.995i)27-s + (0.683 + 0.729i)29-s + (0.707 − 0.707i)31-s + (−0.258 + 0.965i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $-0.224 + 0.974i$
Analytic conductor: \(15.4551\)
Root analytic conductor: \(15.4551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (1051, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3328,\ (0:\ ),\ -0.224 + 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.276366641 + 1.604430427i\)
\(L(\frac12)\) \(\approx\) \(1.276366641 + 1.604430427i\)
\(L(1)\) \(\approx\) \(1.141512501 + 0.5337312062i\)
\(L(1)\) \(\approx\) \(1.141512501 + 0.5337312062i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.0327 + 0.999i)T \)
5 \( 1 + (0.773 - 0.634i)T \)
7 \( 1 + (-0.442 + 0.896i)T \)
11 \( 1 + (0.973 + 0.227i)T \)
17 \( 1 + (0.793 + 0.608i)T \)
19 \( 1 + (0.582 + 0.812i)T \)
23 \( 1 + (-0.321 + 0.946i)T \)
29 \( 1 + (0.683 + 0.729i)T \)
31 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + (0.812 + 0.582i)T \)
41 \( 1 + (-0.321 + 0.946i)T \)
43 \( 1 + (-0.0327 - 0.999i)T \)
47 \( 1 + (-0.923 + 0.382i)T \)
53 \( 1 + (0.290 + 0.956i)T \)
59 \( 1 + (0.352 - 0.935i)T \)
61 \( 1 + (0.528 - 0.849i)T \)
67 \( 1 + (0.0327 - 0.999i)T \)
71 \( 1 + (0.896 + 0.442i)T \)
73 \( 1 + (-0.555 + 0.831i)T \)
79 \( 1 + (0.382 - 0.923i)T \)
83 \( 1 + (-0.0980 - 0.995i)T \)
89 \( 1 + (0.751 + 0.659i)T \)
97 \( 1 + (-0.965 - 0.258i)T \)
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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

−18.63963354431555094409063904688, −17.812416258105875810829396817159, −17.55546152754183199897887065707, −16.63509108523539184378228955145, −16.20825037694758491185427770950, −14.84218503548136596030724693253, −14.29684260573093641070754355109, −13.72094922653960839781635892989, −13.33053644635052097481218150051, −12.38771072568315047826963111068, −11.68620137532955586413339769898, −11.02174132293380554241016552927, −10.13459892331694696221219853411, −9.5765009035768621231493456098, −8.69028019511611656207187998486, −7.79937243081940529901081806860, −6.94570798334812128918074436049, −6.66654954776801498911379319331, −5.950463583548391563203104774936, −5.05218797760022053732630850059, −3.92068903758214432815125177831, −3.00946349441840293808186961382, −2.46037293174973249264989068862, −1.30803356915201516038205075177, −0.69958872918380845814463287869, 1.12647870706070850188892667226, 1.99013033448485227522920179678, 3.05100475770685581695811966259, 3.69777640327105667272863838197, 4.63766376764760194631182943177, 5.366898099130500803438214703767, 5.97704024337842933874374702572, 6.51444692129391800621678882926, 7.99264504309539554710262554006, 8.54542522019980439389318576687, 9.491683710525139838677617684241, 9.64523358946530551957103243255, 10.319605524209136566547743645142, 11.48727632574279512405116713683, 12.01836077531784750634355535817, 12.64168536548389570825162884200, 13.61057023390121408314717912491, 14.31938319928366140616367391303, 14.888799647298051531480583147643, 15.68693235250937803621934664716, 16.317951874799500476738074879662, 16.91361291903705002393083737979, 17.444409313168339466924825321704, 18.28329020015614351657419258508, 19.11985726965265635214012791533

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