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Properties

Label 2-3724-931.284-c0-0-0
Degree $2$
Conductor $3724$
Sign $0.747 + 0.664i$
Analytic cond. $1.85851$
Root an. cond. $1.36327$
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  + (−0.0332 − 0.443i)5-s + (0.826 − 0.563i)7-s + (0.365 − 0.930i)9-s + (0.266 + 0.680i)11-s + (1.07 + 0.997i)17-s + (−0.5 − 0.866i)19-s + (−0.535 + 0.496i)23-s + (0.792 − 0.119i)25-s + (−0.277 − 0.347i)35-s + (0.400 − 0.193i)43-s + (−0.425 − 0.131i)45-s + (−0.147 − 0.0222i)47-s + (0.365 − 0.930i)49-s + (0.292 − 0.141i)55-s + (−1.88 + 0.582i)61-s + ⋯
L(s)  = 1  + (−0.0332 − 0.443i)5-s + (0.826 − 0.563i)7-s + (0.365 − 0.930i)9-s + (0.266 + 0.680i)11-s + (1.07 + 0.997i)17-s + (−0.5 − 0.866i)19-s + (−0.535 + 0.496i)23-s + (0.792 − 0.119i)25-s + (−0.277 − 0.347i)35-s + (0.400 − 0.193i)43-s + (−0.425 − 0.131i)45-s + (−0.147 − 0.0222i)47-s + (0.365 − 0.930i)49-s + (0.292 − 0.141i)55-s + (−1.88 + 0.582i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $0.747 + 0.664i$
Analytic conductor: \(1.85851\)
Root analytic conductor: \(1.36327\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (3077, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3724,\ (\ :0),\ 0.747 + 0.664i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.512326872\)
\(L(\frac12)\) \(\approx\) \(1.512326872\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.826 + 0.563i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (-0.365 + 0.930i)T^{2} \)
5 \( 1 + (0.0332 + 0.443i)T + (-0.988 + 0.149i)T^{2} \)
11 \( 1 + (-0.266 - 0.680i)T + (-0.733 + 0.680i)T^{2} \)
13 \( 1 + (0.222 + 0.974i)T^{2} \)
17 \( 1 + (-1.07 - 0.997i)T + (0.0747 + 0.997i)T^{2} \)
23 \( 1 + (0.535 - 0.496i)T + (0.0747 - 0.997i)T^{2} \)
29 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.826 + 0.563i)T^{2} \)
41 \( 1 + (-0.623 - 0.781i)T^{2} \)
43 \( 1 + (-0.400 + 0.193i)T + (0.623 - 0.781i)T^{2} \)
47 \( 1 + (0.147 + 0.0222i)T + (0.955 + 0.294i)T^{2} \)
53 \( 1 + (-0.826 - 0.563i)T^{2} \)
59 \( 1 + (0.988 + 0.149i)T^{2} \)
61 \( 1 + (1.88 - 0.582i)T + (0.826 - 0.563i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (-0.988 + 0.149i)T + (0.955 - 0.294i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.914 + 1.14i)T + (-0.222 + 0.974i)T^{2} \)
89 \( 1 + (0.733 + 0.680i)T^{2} \)
97 \( 1 - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.633715546364763825359845315180, −7.84222585390330346932986497934, −7.21084063356360609020403936775, −6.45176047556343573913488247769, −5.60078198367316622934951342270, −4.63630952608616652089160598551, −4.15462820170140298998156766416, −3.25733649106893363021871587487, −1.85371954292119635191926526194, −1.03967764171324950056753456484, 1.35142123893313140028802424287, 2.36504792047212276565044186775, 3.20029445940263366437457229945, 4.27656565924368688306536060048, 5.07112857897612103381407635404, 5.72308961546863652239225410149, 6.54297234000860935631023513971, 7.50914130213251536503204438836, 7.988516595416687822933699757925, 8.634626812457710143558474202153

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