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Properties

Label 1-3724-3724.1203-r0-0-0
Degree $1$
Conductor $3724$
Sign $0.695 - 0.718i$
Analytic cond. $17.2941$
Root an. cond. $17.2941$
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  + (−0.411 − 0.911i)3-s + (−0.995 − 0.0995i)5-s + (−0.661 + 0.749i)9-s + (−0.365 − 0.930i)11-s + (0.0249 + 0.999i)13-s + (0.318 + 0.947i)15-s + (−0.921 − 0.388i)17-s + (0.124 − 0.992i)23-s + (0.980 + 0.198i)25-s + (0.955 + 0.294i)27-s + (−0.998 − 0.0498i)29-s + (−0.5 − 0.866i)31-s + (−0.698 + 0.715i)33-s + (−0.222 + 0.974i)37-s + (0.900 − 0.433i)39-s + ⋯
L(s)  = 1  + (−0.411 − 0.911i)3-s + (−0.995 − 0.0995i)5-s + (−0.661 + 0.749i)9-s + (−0.365 − 0.930i)11-s + (0.0249 + 0.999i)13-s + (0.318 + 0.947i)15-s + (−0.921 − 0.388i)17-s + (0.124 − 0.992i)23-s + (0.980 + 0.198i)25-s + (0.955 + 0.294i)27-s + (−0.998 − 0.0498i)29-s + (−0.5 − 0.866i)31-s + (−0.698 + 0.715i)33-s + (−0.222 + 0.974i)37-s + (0.900 − 0.433i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $0.695 - 0.718i$
Analytic conductor: \(17.2941\)
Root analytic conductor: \(17.2941\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (1203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3724,\ (0:\ ),\ 0.695 - 0.718i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5987099441 - 0.2539538053i\)
\(L(\frac12)\) \(\approx\) \(0.5987099441 - 0.2539538053i\)
\(L(1)\) \(\approx\) \(0.6183671921 - 0.1901015515i\)
\(L(1)\) \(\approx\) \(0.6183671921 - 0.1901015515i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.411 - 0.911i)T \)
5 \( 1 + (-0.995 - 0.0995i)T \)
11 \( 1 + (-0.365 - 0.930i)T \)
13 \( 1 + (0.0249 + 0.999i)T \)
17 \( 1 + (-0.921 - 0.388i)T \)
23 \( 1 + (0.124 - 0.992i)T \)
29 \( 1 + (-0.998 - 0.0498i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.222 + 0.974i)T \)
41 \( 1 + (0.411 + 0.911i)T \)
43 \( 1 + (-0.698 + 0.715i)T \)
47 \( 1 + (0.878 + 0.478i)T \)
53 \( 1 + (-0.124 + 0.992i)T \)
59 \( 1 + (0.270 + 0.962i)T \)
61 \( 1 + (-0.542 + 0.840i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 + (-0.542 - 0.840i)T \)
73 \( 1 + (0.0249 - 0.999i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (0.365 - 0.930i)T \)
89 \( 1 + (-0.980 - 0.198i)T \)
97 \( 1 + (-0.173 + 0.984i)T \)
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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

−18.670124925356146102126407560453, −17.78046492746887180969081541097, −17.45122373924504749951889959578, −16.56847852344843392798186576895, −15.61448868904695245977419420042, −15.565622432675032460925122799271, −14.90379699804721250862951020170, −14.130031654515140369096044157516, −12.891945176073041150334401024311, −12.54890350364210686307328489848, −11.62596201834615240592989536850, −11.02492564592435312419408462155, −10.49359642005809462678297143310, −9.760811619581941413418355065205, −8.91557875805787541129615221569, −8.28150805568453653064256723029, −7.33124053789604892742612038291, −6.85003705931689615703071467765, −5.55341054116266166353134989275, −5.23065937978343353383229990861, −4.2050073039466383155469239887, −3.753961183267680760769632872011, −2.9528652933736314106247329511, −1.87120454254662154428252535912, −0.4282329891016105905996140732, 0.46512661999840478290809438929, 1.46631532929285572641133298340, 2.46922737380786565285885948007, 3.2059157261799987317382454581, 4.322158599843516180992946556938, 4.82911374538671188310294484322, 5.966440647633755711755797272290, 6.46089628063705443014719597019, 7.35396734620461046387336237519, 7.80056871715476500797826179762, 8.69615298233376071559808963559, 9.12506457971308888255937133072, 10.52314729853460560529105854148, 11.14843626863462569753715646320, 11.6032666624800465019120439618, 12.19974593813923097594112494678, 13.133692259654448036398732730080, 13.47307997672635051491446605414, 14.38593103966999376401505488273, 15.0999259901292803874032100721, 15.98709514824554815794794262454, 16.63360835224320606383947337958, 16.92189266196811352751789291665, 18.17952913075841510169457706609, 18.53283290952662075984985487140

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