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Properties

Label 1-3744-3744.355-r1-0-0
Degree $1$
Conductor $3744$
Sign $0.861 - 0.507i$
Analytic cond. $402.348$
Root an. cond. $402.348$
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  + (−0.965 − 0.258i)5-s + (−0.866 + 0.5i)7-s + (0.707 − 0.707i)11-s + (0.5 − 0.866i)17-s + (0.965 − 0.258i)19-s + (0.866 + 0.5i)23-s + (0.866 + 0.5i)25-s + (0.707 + 0.707i)29-s + (−0.5 − 0.866i)31-s + (0.965 − 0.258i)35-s + (−0.965 − 0.258i)37-s + (0.866 + 0.5i)41-s + (0.965 + 0.258i)43-s + (0.5 − 0.866i)47-s + (0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)5-s + (−0.866 + 0.5i)7-s + (0.707 − 0.707i)11-s + (0.5 − 0.866i)17-s + (0.965 − 0.258i)19-s + (0.866 + 0.5i)23-s + (0.866 + 0.5i)25-s + (0.707 + 0.707i)29-s + (−0.5 − 0.866i)31-s + (0.965 − 0.258i)35-s + (−0.965 − 0.258i)37-s + (0.866 + 0.5i)41-s + (0.965 + 0.258i)43-s + (0.5 − 0.866i)47-s + (0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.861 - 0.507i$
Analytic conductor: \(402.348\)
Root analytic conductor: \(402.348\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (355, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3744,\ (1:\ ),\ 0.861 - 0.507i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.841617727 - 0.5023460608i\)
\(L(\frac12)\) \(\approx\) \(1.841617727 - 0.5023460608i\)
\(L(1)\) \(\approx\) \(0.9537001661 - 0.08227549604i\)
\(L(1)\) \(\approx\) \(0.9537001661 - 0.08227549604i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (-0.965 - 0.258i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.965 - 0.258i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.707 + 0.707i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.965 - 0.258i)T \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (0.965 + 0.258i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.707 - 0.707i)T \)
59 \( 1 + (-0.707 + 0.707i)T \)
61 \( 1 + (0.258 - 0.965i)T \)
67 \( 1 + (-0.965 + 0.258i)T \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.965 - 0.258i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−18.723319200186003195577981421968, −17.73678979771472470442777362702, −17.08738574773325513043766077389, −16.40382054935488400089540315541, −15.76043855368027421100110155508, −15.177507117101672937326791426135, −14.361264323105509438599439111319, −13.861366775534777198301037594413, −12.70816815230081010775869151098, −12.37343684256954662222233761966, −11.71915549777370564448595938153, −10.692635434807022743382233742981, −10.33871214471945218618122625322, −9.37822995299957487785383551343, −8.77489822267139075041086876861, −7.74297234665644123507283294247, −7.24405166827419306889735543959, −6.61088854926457002523857467275, −5.80722279699742342725682496040, −4.6973995951704475615095468316, −4.00800795274904518101436299986, −3.413537501348974542127412358477, −2.666971356101087630911706710121, −1.35887532993345139108467802898, −0.61276198309529290889651794263, 0.54259414172366603886531905048, 1.08033727843568700566819458300, 2.52269103024180613472887112846, 3.34025695681710226807694126308, 3.702606663880341444108533057787, 4.859215193112895508110410877853, 5.4884602795933871881442099908, 6.36399414942357830322716782213, 7.19700772370258753144614890833, 7.68375789617549264529482046164, 8.88633436473377642255267225229, 9.03385820153756450204747179704, 9.905910872378731515970497725496, 10.93383168664493768955615077204, 11.59294000995167126874802170213, 12.07785226025399159522510056964, 12.775481743342917913227920999367, 13.558772910032955126261699088390, 14.2764751218837504007534849892, 15.09301268417551100262412659378, 15.781078821772696000810309095676, 16.29001690355742579563395371653, 16.745051427720531282932377605831, 17.75954273508762511575501861066, 18.60909589491686428657274729456

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