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Properties

Label 1-3724-3724.23-r1-0-0
Degree $1$
Conductor $3724$
Sign $0.677 - 0.735i$
Analytic cond. $400.199$
Root an. cond. $400.199$
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  + (0.583 − 0.811i)3-s + (0.995 + 0.0995i)5-s + (−0.318 − 0.947i)9-s + (0.988 + 0.149i)11-s + (−0.853 + 0.521i)13-s + (0.661 − 0.749i)15-s + (−0.124 − 0.992i)17-s + (0.797 + 0.603i)23-s + (0.980 + 0.198i)25-s + (−0.955 − 0.294i)27-s + (0.456 + 0.889i)29-s − 31-s + (0.698 − 0.715i)33-s + (0.955 − 0.294i)37-s + (−0.0747 + 0.997i)39-s + ⋯
L(s)  = 1  + (0.583 − 0.811i)3-s + (0.995 + 0.0995i)5-s + (−0.318 − 0.947i)9-s + (0.988 + 0.149i)11-s + (−0.853 + 0.521i)13-s + (0.661 − 0.749i)15-s + (−0.124 − 0.992i)17-s + (0.797 + 0.603i)23-s + (0.980 + 0.198i)25-s + (−0.955 − 0.294i)27-s + (0.456 + 0.889i)29-s − 31-s + (0.698 − 0.715i)33-s + (0.955 − 0.294i)37-s + (−0.0747 + 0.997i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $0.677 - 0.735i$
Analytic conductor: \(400.199\)
Root analytic conductor: \(400.199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3724,\ (1:\ ),\ 0.677 - 0.735i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.874595589 - 1.698209109i\)
\(L(\frac12)\) \(\approx\) \(3.874595589 - 1.698209109i\)
\(L(1)\) \(\approx\) \(1.617525228 - 0.4301265551i\)
\(L(1)\) \(\approx\) \(1.617525228 - 0.4301265551i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.583 - 0.811i)T \)
5 \( 1 + (0.995 + 0.0995i)T \)
11 \( 1 + (0.988 + 0.149i)T \)
13 \( 1 + (-0.853 + 0.521i)T \)
17 \( 1 + (-0.124 - 0.992i)T \)
23 \( 1 + (0.797 + 0.603i)T \)
29 \( 1 + (0.456 + 0.889i)T \)
31 \( 1 - T \)
37 \( 1 + (0.955 - 0.294i)T \)
41 \( 1 + (0.995 + 0.0995i)T \)
43 \( 1 + (-0.270 - 0.962i)T \)
47 \( 1 + (0.853 - 0.521i)T \)
53 \( 1 + (-0.124 + 0.992i)T \)
59 \( 1 + (-0.698 + 0.715i)T \)
61 \( 1 + (0.456 + 0.889i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 + (0.998 - 0.0498i)T \)
73 \( 1 + (0.878 - 0.478i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.365 + 0.930i)T \)
89 \( 1 + (-0.661 + 0.749i)T \)
97 \( 1 + (0.766 + 0.642i)T \)
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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

−18.62710622354680627914203997531, −17.62317772442421745060591635801, −17.045558339184284699486675286706, −16.70311941364945389716601524704, −15.74015543482971900086810939958, −14.93558185346039162398475376427, −14.45917858618740052990414923550, −14.00885155348844690899686805377, −12.91927589751391844165043607064, −12.686104191926938298151977774325, −11.36692187412678582536179933526, −10.830470411994671242603442615658, −9.891763041870851158594347116299, −9.65013440543026193065271918344, −8.82386698112225748264301323007, −8.24308253802313868404774606038, −7.2956264318444419808806174071, −6.322347237865624490338216453864, −5.72050071595094357999086081356, −4.819231336022499471937247536273, −4.23647837213703875416805237734, −3.28191842991342713335465659759, −2.52559260452354887528599069268, −1.812672734730605453833423976923, −0.74242598817262223406133511814, 0.7221164998286348730077993801, 1.46420076067320334148332955287, 2.25484708736051399732261557358, 2.85189895293457043472605345680, 3.801449267556588282538879546014, 4.823407376159014756328794609418, 5.645783828276927299844553230020, 6.46349164798295357237863502700, 7.12583280927967744562897047883, 7.499261144580688344235255917487, 8.851154327970579060151279904711, 9.18706616885401317246490289956, 9.67150133566221713507159695899, 10.74485992399359602619629594880, 11.58716635855127587129874860672, 12.28969787003522511215070240581, 12.871611741618286788121353636192, 13.73612055937034391678847586462, 14.12672708481188606404132494415, 14.6858524350815376001030410202, 15.401336294902477632907431131104, 16.696115058243338903503040242131, 16.9497500479182787541737468582, 17.89476839293976794787119249136, 18.24521019531505061305829931689

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