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Properties

Label 1-3723-3723.1082-r1-0-0
Degree $1$
Conductor $3723$
Sign $0.667 + 0.744i$
Analytic cond. $400.091$
Root an. cond. $400.091$
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  + (0.422 + 0.906i)2-s + (−0.642 + 0.766i)4-s + (0.999 + 0.0436i)5-s + (−0.608 + 0.793i)7-s + (−0.965 − 0.258i)8-s + (0.382 + 0.923i)10-s + (0.675 + 0.737i)11-s + (−0.819 − 0.573i)13-s + (−0.976 − 0.216i)14-s + (−0.173 − 0.984i)16-s + (0.906 − 0.422i)19-s + (−0.675 + 0.737i)20-s + (−0.382 + 0.923i)22-s + (0.0436 − 0.999i)23-s + (0.996 + 0.0871i)25-s + (0.173 − 0.984i)26-s + ⋯
L(s)  = 1  + (0.422 + 0.906i)2-s + (−0.642 + 0.766i)4-s + (0.999 + 0.0436i)5-s + (−0.608 + 0.793i)7-s + (−0.965 − 0.258i)8-s + (0.382 + 0.923i)10-s + (0.675 + 0.737i)11-s + (−0.819 − 0.573i)13-s + (−0.976 − 0.216i)14-s + (−0.173 − 0.984i)16-s + (0.906 − 0.422i)19-s + (−0.675 + 0.737i)20-s + (−0.382 + 0.923i)22-s + (0.0436 − 0.999i)23-s + (0.996 + 0.0871i)25-s + (0.173 − 0.984i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3723 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3723 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3723\)    =    \(3 \cdot 17 \cdot 73\)
Sign: $0.667 + 0.744i$
Analytic conductor: \(400.091\)
Root analytic conductor: \(400.091\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3723} (1082, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3723,\ (1:\ ),\ 0.667 + 0.744i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.241763710 + 1.001385574i\)
\(L(\frac12)\) \(\approx\) \(2.241763710 + 1.001385574i\)
\(L(1)\) \(\approx\) \(1.101909248 + 0.6959907418i\)
\(L(1)\) \(\approx\) \(1.101909248 + 0.6959907418i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
73 \( 1 \)
good2 \( 1 + (0.422 + 0.906i)T \)
5 \( 1 + (0.999 + 0.0436i)T \)
7 \( 1 + (-0.608 + 0.793i)T \)
11 \( 1 + (0.675 + 0.737i)T \)
13 \( 1 + (-0.819 - 0.573i)T \)
19 \( 1 + (0.906 - 0.422i)T \)
23 \( 1 + (0.0436 - 0.999i)T \)
29 \( 1 + (-0.675 + 0.737i)T \)
31 \( 1 + (-0.953 + 0.300i)T \)
37 \( 1 + (0.737 - 0.675i)T \)
41 \( 1 + (-0.843 - 0.537i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.573 + 0.819i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (-0.984 + 0.173i)T \)
61 \( 1 + (0.537 - 0.843i)T \)
67 \( 1 + (-0.984 + 0.173i)T \)
71 \( 1 + (0.999 - 0.0436i)T \)
79 \( 1 + (-0.976 - 0.216i)T \)
83 \( 1 + T \)
89 \( 1 + (0.984 - 0.173i)T \)
97 \( 1 + (-0.130 + 0.991i)T \)
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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

−18.64575049405434881627159401519, −17.71455847764109539477199411100, −16.92831438744228916875739217157, −16.65319474832323921287863061458, −15.491097092162700971238852910056, −14.55523538957605192777371238508, −14.04205719065543209009582613198, −13.48781593455609757542851285652, −13.04436808849033954012151586079, −12.05741635773620043694595774968, −11.52454582333807824962531028325, −10.69313029070149152728984670995, −9.98782602650268253395915723471, −9.42658068716400213557890534005, −9.070342717083251694086850602543, −7.76092565701499677455255212632, −6.8610680910605023075508866922, −6.070768441551993933644013951984, −5.5120963186599300943467611474, −4.634233948889697911858673666942, −3.74235871880476195168876689425, −3.21140887481242748536041762820, −2.229117025607091569373329428094, −1.45250394921905801376030935570, −0.71541801041567170257700338426, 0.39406877812640164806228451294, 1.760692153800785506947667846106, 2.69726809742362399444959832840, 3.273505952974794444650189824079, 4.436656040294700147815740662284, 5.18208321448117060800752911008, 5.69958370852494057818538558356, 6.48133045359439908267343773250, 7.04598826655968486658563676668, 7.79786790971859104086751573371, 8.922644255148972922161703465852, 9.33470735404647529344796004903, 9.818329185870795286892428839674, 10.87149215174003778275197626935, 12.00261849739588501840758894435, 12.66500279808822484561922858267, 12.89237825174445420886095096406, 13.91754516721911878738177515048, 14.57894768975610621945199389955, 14.93390240385088758373609668728, 15.82782264848275301191080450477, 16.47239712102988022257197193397, 17.12097921362210519769437085137, 17.80965263420157115720587310171, 18.22643955695150664147699940930

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