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Properties

Label 2-3784-3784.1171-c0-0-0
Degree $2$
Conductor $3784$
Sign $0.635 + 0.772i$
Analytic cond. $1.88846$
Root an. cond. $1.37421$
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  + (0.858 − 0.512i)2-s + (0.0518 + 0.263i)3-s + (0.473 − 0.880i)4-s + (0.179 + 0.199i)6-s + (−0.0448 − 0.998i)8-s + (0.858 − 0.351i)9-s + (0.193 + 0.981i)11-s + (0.256 + 0.0791i)12-s + (−0.550 − 0.834i)16-s + (−0.470 + 0.940i)17-s + (0.556 − 0.742i)18-s + (1.29 − 1.52i)19-s + (0.669 + 0.743i)22-s + (0.260 − 0.0636i)24-s + (−0.946 − 0.323i)25-s + ⋯
L(s)  = 1  + (0.858 − 0.512i)2-s + (0.0518 + 0.263i)3-s + (0.473 − 0.880i)4-s + (0.179 + 0.199i)6-s + (−0.0448 − 0.998i)8-s + (0.858 − 0.351i)9-s + (0.193 + 0.981i)11-s + (0.256 + 0.0791i)12-s + (−0.550 − 0.834i)16-s + (−0.470 + 0.940i)17-s + (0.556 − 0.742i)18-s + (1.29 − 1.52i)19-s + (0.669 + 0.743i)22-s + (0.260 − 0.0636i)24-s + (−0.946 − 0.323i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3784\)    =    \(2^{3} \cdot 11 \cdot 43\)
Sign: $0.635 + 0.772i$
Analytic conductor: \(1.88846\)
Root analytic conductor: \(1.37421\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3784} (1171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3784,\ (\ :0),\ 0.635 + 0.772i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.382118096\)
\(L(\frac12)\) \(\approx\) \(2.382118096\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.858 + 0.512i)T \)
11 \( 1 + (-0.193 - 0.981i)T \)
43 \( 1 + (-0.420 - 0.907i)T \)
good3 \( 1 + (-0.0518 - 0.263i)T + (-0.925 + 0.379i)T^{2} \)
5 \( 1 + (0.946 + 0.323i)T^{2} \)
7 \( 1 + (-0.669 - 0.743i)T^{2} \)
13 \( 1 + (-0.992 - 0.119i)T^{2} \)
17 \( 1 + (0.470 - 0.940i)T + (-0.599 - 0.800i)T^{2} \)
19 \( 1 + (-1.29 + 1.52i)T + (-0.163 - 0.986i)T^{2} \)
23 \( 1 + (-0.365 + 0.930i)T^{2} \)
29 \( 1 + (0.925 + 0.379i)T^{2} \)
31 \( 1 + (0.999 - 0.0299i)T^{2} \)
37 \( 1 + (0.978 - 0.207i)T^{2} \)
41 \( 1 + (-0.607 + 1.42i)T + (-0.691 - 0.722i)T^{2} \)
47 \( 1 + (-0.936 - 0.351i)T^{2} \)
53 \( 1 + (-0.193 + 0.981i)T^{2} \)
59 \( 1 + (0.0890 - 1.98i)T + (-0.995 - 0.0896i)T^{2} \)
61 \( 1 + (0.999 + 0.0299i)T^{2} \)
67 \( 1 + (0.0156 - 0.208i)T + (-0.988 - 0.149i)T^{2} \)
71 \( 1 + (-0.251 + 0.967i)T^{2} \)
73 \( 1 + (0.492 + 1.89i)T + (-0.873 + 0.486i)T^{2} \)
79 \( 1 + (0.104 + 0.994i)T^{2} \)
83 \( 1 + (1.49 - 0.836i)T + (0.525 - 0.850i)T^{2} \)
89 \( 1 + (1.46 + 1.35i)T + (0.0747 + 0.997i)T^{2} \)
97 \( 1 + (-0.112 + 0.833i)T + (-0.963 - 0.266i)T^{2} \)
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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

−8.888202691562550171622210304706, −7.43277039409463119928186389014, −7.14934152036317885037365295975, −6.21763511113718750083730355883, −5.46454118726153850379802520137, −4.36984340668981035804892443523, −4.28529442646546700119200020542, −3.14946894133317406338871861595, −2.20906860680838934457705628632, −1.21661902787763290125618463602, 1.44130187651474095221300513352, 2.57458558166132987180330246160, 3.53336331062227905124155724198, 4.15955118015058229334500899483, 5.18016216805766680847525533862, 5.72812136329656610775768843258, 6.54782027508218952243676083869, 7.29002403943102221660452050337, 7.83778571038937225633963040027, 8.465776877649414580006750046497

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