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Properties

Label 1-3724-3724.1443-r0-0-0
Degree $1$
Conductor $3724$
Sign $-0.572 + 0.820i$
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  + (−0.900 − 0.433i)3-s + (−0.900 − 0.433i)5-s + (0.623 + 0.781i)9-s + (−0.623 + 0.781i)11-s + (−0.623 + 0.781i)13-s + (0.623 + 0.781i)15-s + (−0.222 + 0.974i)17-s + (0.222 + 0.974i)23-s + (0.623 + 0.781i)25-s + (−0.222 − 0.974i)27-s + (0.222 − 0.974i)29-s + 31-s + (0.900 − 0.433i)33-s + (0.222 − 0.974i)37-s + (0.900 − 0.433i)39-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)3-s + (−0.900 − 0.433i)5-s + (0.623 + 0.781i)9-s + (−0.623 + 0.781i)11-s + (−0.623 + 0.781i)13-s + (0.623 + 0.781i)15-s + (−0.222 + 0.974i)17-s + (0.222 + 0.974i)23-s + (0.623 + 0.781i)25-s + (−0.222 − 0.974i)27-s + (0.222 − 0.974i)29-s + 31-s + (0.900 − 0.433i)33-s + (0.222 − 0.974i)37-s + (0.900 − 0.433i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.572 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.572 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $-0.572 + 0.820i$
Analytic conductor: \(17.2941\)
Root analytic conductor: \(17.2941\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (1443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3724,\ (0:\ ),\ -0.572 + 0.820i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2207852108 + 0.4232039139i\)
\(L(\frac12)\) \(\approx\) \(0.2207852108 + 0.4232039139i\)
\(L(1)\) \(\approx\) \(0.6054212624 + 0.03396219659i\)
\(L(1)\) \(\approx\) \(0.6054212624 + 0.03396219659i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (-0.900 - 0.433i)T \)
11 \( 1 + (-0.623 + 0.781i)T \)
13 \( 1 + (-0.623 + 0.781i)T \)
17 \( 1 + (-0.222 + 0.974i)T \)
23 \( 1 + (0.222 + 0.974i)T \)
29 \( 1 + (0.222 - 0.974i)T \)
31 \( 1 + T \)
37 \( 1 + (0.222 - 0.974i)T \)
41 \( 1 + (0.900 + 0.433i)T \)
43 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + (-0.623 + 0.781i)T \)
53 \( 1 + (0.222 + 0.974i)T \)
59 \( 1 + (-0.900 + 0.433i)T \)
61 \( 1 + (-0.222 + 0.974i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.222 - 0.974i)T \)
73 \( 1 + (0.623 + 0.781i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.623 - 0.781i)T \)
89 \( 1 + (-0.623 - 0.781i)T \)
97 \( 1 - T \)
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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

−18.34064840684966617377075138734, −17.78702290384334653909692471353, −16.91378339098471971929039529338, −16.2036404069389811137100803032, −15.80389771526581288744284300315, −15.128597594554595530475138393442, −14.45900296063897878303604659203, −13.5502797685336922964724032833, −12.54397292666314628065621844092, −12.208709964797145441282872820417, −11.219718529415009648311170634705, −10.94474977469769086302458877908, −10.20758556810138028344102682763, −9.4855099532607850615567413334, −8.41900845139677221430076780909, −7.84410471438746657695760991946, −6.905912688260728580363936415067, −6.43899214677386265188241946023, −5.33242794072439894630775784162, −4.9017394118127830042019810077, −4.07070883668543425746925358705, −3.12139599269464001165473338728, −2.62156819825335523311508126716, −0.9153049818618977124754135400, −0.231504777912273547377087533973, 0.97547168057768310591344503749, 1.869035928637628376393359908400, 2.74716590426761895181879570216, 4.22523042036935968707697069134, 4.35729601918378196329001094570, 5.30930797777658886289508094790, 6.0641632131346695812936646379, 6.93674957107317913534228248539, 7.617097286123619147083579336513, 8.007627504742951095304877933507, 9.13102915045435981983355236716, 9.858297230909980890078322144604, 10.74370305206275552150363139660, 11.30890782259891555124058920878, 12.08430234504247065658751624008, 12.47146952272945266710230732326, 13.12672844411389441860813381101, 13.928090834983309877988028412923, 14.98982172540971513086003735801, 15.55210919289968286189745316116, 16.127218354670329120670570420401, 16.95706657336603667460452801961, 17.40263545773381172315204871789, 18.06298843423469916208738039102, 19.02771114870954116787204404122

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