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Properties

Label 2-3724-931.303-c0-0-0
Degree $2$
Conductor $3724$
Sign $0.656 - 0.754i$
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  + (0.603 + 1.53i)5-s + (0.623 − 0.781i)7-s + (0.955 + 0.294i)9-s + (−0.425 + 0.131i)11-s + (0.123 + 0.0841i)17-s + (−0.5 + 0.866i)19-s + (1.57 − 1.07i)23-s + (−1.26 + 1.17i)25-s + (1.57 + 0.487i)35-s + (0.0931 + 0.116i)43-s + (0.123 + 1.64i)45-s + (−0.914 − 0.848i)47-s + (−0.222 − 0.974i)49-s + (−0.458 − 0.574i)55-s + (−0.0332 + 0.443i)61-s + ⋯
L(s)  = 1  + (0.603 + 1.53i)5-s + (0.623 − 0.781i)7-s + (0.955 + 0.294i)9-s + (−0.425 + 0.131i)11-s + (0.123 + 0.0841i)17-s + (−0.5 + 0.866i)19-s + (1.57 − 1.07i)23-s + (−1.26 + 1.17i)25-s + (1.57 + 0.487i)35-s + (0.0931 + 0.116i)43-s + (0.123 + 1.64i)45-s + (−0.914 − 0.848i)47-s + (−0.222 − 0.974i)49-s + (−0.458 − 0.574i)55-s + (−0.0332 + 0.443i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $0.656 - 0.754i$
Analytic conductor: \(1.85851\)
Root analytic conductor: \(1.36327\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (2165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3724,\ (\ :0),\ 0.656 - 0.754i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.654771431\)
\(L(\frac12)\) \(\approx\) \(1.654771431\)
\(L(1)\) 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 \( 1 + (-0.623 + 0.781i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (-0.955 - 0.294i)T^{2} \)
5 \( 1 + (-0.603 - 1.53i)T + (-0.733 + 0.680i)T^{2} \)
11 \( 1 + (0.425 - 0.131i)T + (0.826 - 0.563i)T^{2} \)
13 \( 1 + (0.900 + 0.433i)T^{2} \)
17 \( 1 + (-0.123 - 0.0841i)T + (0.365 + 0.930i)T^{2} \)
23 \( 1 + (-1.57 + 1.07i)T + (0.365 - 0.930i)T^{2} \)
29 \( 1 + (-0.623 + 0.781i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.988 + 0.149i)T^{2} \)
41 \( 1 + (0.222 + 0.974i)T^{2} \)
43 \( 1 + (-0.0931 - 0.116i)T + (-0.222 + 0.974i)T^{2} \)
47 \( 1 + (0.914 + 0.848i)T + (0.0747 + 0.997i)T^{2} \)
53 \( 1 + (0.988 - 0.149i)T^{2} \)
59 \( 1 + (0.733 + 0.680i)T^{2} \)
61 \( 1 + (0.0332 - 0.443i)T + (-0.988 - 0.149i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (1.46 - 1.36i)T + (0.0747 - 0.997i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.367 + 1.61i)T + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (-0.826 - 0.563i)T^{2} \)
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

−8.706996474497478211108880191763, −7.87843078269504146542199941947, −7.13496007402907942140246814633, −6.83764896181333719506292232122, −5.94978910831408655270146044595, −4.96789562802720036978669261422, −4.19950395420115053947519867775, −3.26298836567989615144099578225, −2.35737654000609618325822366356, −1.45863356054543155885098475401, 1.10742547221934459963893636076, 1.84107507055766738721045278617, 2.97082886842707629633200631578, 4.28485815979488039438497845001, 4.95228427452076211922759557950, 5.34994658838717419794732524289, 6.24162471419653995058909460598, 7.20217344598653194322042380950, 8.011547563413084188202837539133, 8.729542292804206758061878235811

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