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Properties

Label 1-4056-4056.1091-r0-0-0
Degree $1$
Conductor $4056$
Sign $0.956 + 0.293i$
Analytic cond. $18.8359$
Root an. cond. $18.8359$
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  + (0.748 + 0.663i)5-s + (−0.970 − 0.239i)7-s + (0.568 + 0.822i)11-s + (0.970 + 0.239i)17-s − 19-s + 23-s + (0.120 + 0.992i)25-s + (0.568 − 0.822i)29-s + (0.120 − 0.992i)31-s + (−0.568 − 0.822i)35-s + (0.120 − 0.992i)37-s + (0.885 − 0.464i)41-s + (0.120 + 0.992i)43-s + (0.354 − 0.935i)47-s + (0.885 + 0.464i)49-s + ⋯
L(s)  = 1  + (0.748 + 0.663i)5-s + (−0.970 − 0.239i)7-s + (0.568 + 0.822i)11-s + (0.970 + 0.239i)17-s − 19-s + 23-s + (0.120 + 0.992i)25-s + (0.568 − 0.822i)29-s + (0.120 − 0.992i)31-s + (−0.568 − 0.822i)35-s + (0.120 − 0.992i)37-s + (0.885 − 0.464i)41-s + (0.120 + 0.992i)43-s + (0.354 − 0.935i)47-s + (0.885 + 0.464i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4056\)    =    \(2^{3} \cdot 3 \cdot 13^{2}\)
Sign: $0.956 + 0.293i$
Analytic conductor: \(18.8359\)
Root analytic conductor: \(18.8359\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4056} (1091, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4056,\ (0:\ ),\ 0.956 + 0.293i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.943985371 + 0.2912489643i\)
\(L(\frac12)\) \(\approx\) \(1.943985371 + 0.2912489643i\)
\(L(1)\) \(\approx\) \(1.194809621 + 0.1293580503i\)
\(L(1)\) \(\approx\) \(1.194809621 + 0.1293580503i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (0.748 + 0.663i)T \)
7 \( 1 + (-0.970 - 0.239i)T \)
11 \( 1 + (0.568 + 0.822i)T \)
17 \( 1 + (0.970 + 0.239i)T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 + (0.568 - 0.822i)T \)
31 \( 1 + (0.120 - 0.992i)T \)
37 \( 1 + (0.120 - 0.992i)T \)
41 \( 1 + (0.885 - 0.464i)T \)
43 \( 1 + (0.120 + 0.992i)T \)
47 \( 1 + (0.354 - 0.935i)T \)
53 \( 1 + (-0.970 - 0.239i)T \)
59 \( 1 + (-0.748 - 0.663i)T \)
61 \( 1 + (0.970 - 0.239i)T \)
67 \( 1 + (0.354 - 0.935i)T \)
71 \( 1 + (-0.885 + 0.464i)T \)
73 \( 1 + (-0.568 - 0.822i)T \)
79 \( 1 + (0.354 - 0.935i)T \)
83 \( 1 + (0.885 + 0.464i)T \)
89 \( 1 + T \)
97 \( 1 + (0.748 - 0.663i)T \)
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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

−18.58065682233065147085458317256, −17.54734969147986596845023397097, −17.04534641280864998394470168755, −16.35356013871554802198780857893, −16.01108191646955754340177335719, −14.97200898531781999730920032453, −14.220857294812633098043035233835, −13.66267704233938910943660747496, −12.832901423906785288314996725499, −12.48817912674618764121101512903, −11.68817244644285795031473928949, −10.67934031524237470082924060518, −10.13466192079844714349426648981, −9.25586599979072324455664245159, −8.89497557158660060639449939354, −8.1831760295669550151891139069, −7.02528189384153179845039777875, −6.37915243548188618894700638157, −5.79249828051936954700706687326, −5.065082188416381584354077826702, −4.19539405970191020999931871553, −3.18069670654151281413167283187, −2.70194386297507645522028900134, −1.45774188772505591027663296450, −0.81706779754404068861499853173, 0.74782764032226994076230861276, 1.88936063360621277505682171621, 2.55687477473917117241605973879, 3.41816500753405463656460059941, 4.11527894207922750058854932529, 5.06734242364520320137368732600, 6.15906728544268517066763456671, 6.33604857269131750064890736499, 7.22595826721994868079570338053, 7.84915714316270762030517424601, 9.113771593218453927673070035371, 9.48296312913615964006021484798, 10.21400582649805113046538951401, 10.71203111671920036401883998915, 11.620369061632797420154471548454, 12.53139106465792200565050301918, 12.97789709111135513833887330134, 13.69831506678442198848267388453, 14.558184727175764991337429868416, 14.88751312439679878964917724107, 15.75179119866511782712856007380, 16.59979652884714911163132250861, 17.25784132909247973807449159260, 17.58143887504519355728878318427, 18.66076849176881365802230631281

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