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Properties

Label 1-3328-3328.1107-r0-0-0
Degree $1$
Conductor $3328$
Sign $0.212 + 0.977i$
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  + (0.528 + 0.849i)3-s + (−0.634 + 0.773i)5-s + (−0.997 − 0.0654i)7-s + (−0.442 + 0.896i)9-s + (0.729 − 0.683i)11-s + (−0.991 − 0.130i)15-s + (0.130 + 0.991i)17-s + (0.910 − 0.412i)19-s + (−0.471 − 0.881i)21-s + (0.659 − 0.751i)23-s + (−0.195 − 0.980i)25-s + (−0.995 + 0.0980i)27-s + (−0.227 + 0.973i)29-s + (0.707 + 0.707i)31-s + (0.965 + 0.258i)33-s + ⋯
L(s)  = 1  + (0.528 + 0.849i)3-s + (−0.634 + 0.773i)5-s + (−0.997 − 0.0654i)7-s + (−0.442 + 0.896i)9-s + (0.729 − 0.683i)11-s + (−0.991 − 0.130i)15-s + (0.130 + 0.991i)17-s + (0.910 − 0.412i)19-s + (−0.471 − 0.881i)21-s + (0.659 − 0.751i)23-s + (−0.195 − 0.980i)25-s + (−0.995 + 0.0980i)27-s + (−0.227 + 0.973i)29-s + (0.707 + 0.707i)31-s + (0.965 + 0.258i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.212 + 0.977i$
Analytic conductor: \(15.4551\)
Root analytic conductor: \(15.4551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (1107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3328,\ (0:\ ),\ 0.212 + 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.314683854 + 1.059700650i\)
\(L(\frac12)\) \(\approx\) \(1.314683854 + 1.059700650i\)
\(L(1)\) \(\approx\) \(1.021545047 + 0.4366734849i\)
\(L(1)\) \(\approx\) \(1.021545047 + 0.4366734849i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.528 + 0.849i)T \)
5 \( 1 + (-0.634 + 0.773i)T \)
7 \( 1 + (-0.997 - 0.0654i)T \)
11 \( 1 + (0.729 - 0.683i)T \)
17 \( 1 + (0.130 + 0.991i)T \)
19 \( 1 + (0.910 - 0.412i)T \)
23 \( 1 + (0.659 - 0.751i)T \)
29 \( 1 + (-0.227 + 0.973i)T \)
31 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (0.412 - 0.910i)T \)
41 \( 1 + (0.659 - 0.751i)T \)
43 \( 1 + (0.528 - 0.849i)T \)
47 \( 1 + (-0.923 - 0.382i)T \)
53 \( 1 + (0.956 + 0.290i)T \)
59 \( 1 + (0.162 - 0.986i)T \)
61 \( 1 + (0.0327 + 0.999i)T \)
67 \( 1 + (-0.528 - 0.849i)T \)
71 \( 1 + (0.0654 - 0.997i)T \)
73 \( 1 + (0.555 + 0.831i)T \)
79 \( 1 + (0.382 + 0.923i)T \)
83 \( 1 + (0.995 + 0.0980i)T \)
89 \( 1 + (0.946 + 0.321i)T \)
97 \( 1 + (0.258 - 0.965i)T \)
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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

−18.97771808076857988657259510822, −18.07110040160535185742728878488, −17.35732042660490981921633621573, −16.59489376987085818222955450672, −15.97107110377615682414257972858, −15.19433929622282554868755490441, −14.59463921183689376860416117356, −13.46844828035748415245357227383, −13.29719040158553459093971463179, −12.40706701847643296876058002893, −11.81213102078852546117561689736, −11.44575190755653908291430329025, −9.72193221451296376036588786455, −9.57260319908479774294190408511, −8.8308617533300821640102526192, −7.80138598870775267258481985979, −7.47721043548353132633160822889, −6.60380606777351303610233238678, −5.91152131667235994612421622719, −4.88275846369130185024135239407, −3.99123296060952655187164084736, −3.25168998314479486395212882904, −2.519497466247847166725011314989, −1.33995169005495122388053295048, −0.714412991883991086145066432530, 0.76815720492117363297848408306, 2.27881846452098186418145366625, 3.12619497753257422221388771553, 3.55991489739110111951147702704, 4.16173600600196671728328185781, 5.23411837322940976983768959912, 6.139174654354085819768434629004, 6.86185144522211083479426289508, 7.59209374109114915386967794574, 8.5774955172338139878234840203, 9.03939682897435984127743888229, 9.865783925104577166420342266152, 10.656632930151795824083077365998, 10.98405899819184570739524111972, 11.97179764005868930311945681504, 12.70561069707915193874402934027, 13.705003625003248285750031334552, 14.212083218620328685292881632992, 14.92911965982072605348305078042, 15.48706860826418056943522933056, 16.28364088173290455564558561751, 16.553732464708222173011979747488, 17.56262850650218644735856671766, 18.54363952480532658884838920300, 19.26778995350492514614644920168

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