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Properties

Label 2-3784-3784.427-c0-0-0
Degree $2$
Conductor $3784$
Sign $-0.490 - 0.871i$
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  + (−0.691 + 0.722i)2-s + (−0.579 − 0.937i)3-s + (−0.0448 − 0.998i)4-s + (1.07 + 0.229i)6-s + (0.753 + 0.657i)8-s + (−0.0957 + 0.191i)9-s + (0.525 + 0.850i)11-s + (−0.910 + 0.620i)12-s + (−0.995 + 0.0896i)16-s + (1.02 + 1.45i)17-s + (−0.0721 − 0.201i)18-s + (−1.20 + 1.18i)19-s + (−0.978 − 0.207i)22-s + (0.180 − 1.08i)24-s + (−0.999 + 0.0299i)25-s + ⋯
L(s)  = 1  + (−0.691 + 0.722i)2-s + (−0.579 − 0.937i)3-s + (−0.0448 − 0.998i)4-s + (1.07 + 0.229i)6-s + (0.753 + 0.657i)8-s + (−0.0957 + 0.191i)9-s + (0.525 + 0.850i)11-s + (−0.910 + 0.620i)12-s + (−0.995 + 0.0896i)16-s + (1.02 + 1.45i)17-s + (−0.0721 − 0.201i)18-s + (−1.20 + 1.18i)19-s + (−0.978 − 0.207i)22-s + (0.180 − 1.08i)24-s + (−0.999 + 0.0299i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.490 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.490 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3784\)    =    \(2^{3} \cdot 11 \cdot 43\)
Sign: $-0.490 - 0.871i$
Analytic conductor: \(1.88846\)
Root analytic conductor: \(1.37421\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3784} (427, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3784,\ (\ :0),\ -0.490 - 0.871i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3886690378\)
\(L(\frac12)\) \(\approx\) \(0.3886690378\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.691 - 0.722i)T \)
11 \( 1 + (-0.525 - 0.850i)T \)
43 \( 1 + (0.925 - 0.379i)T \)
good3 \( 1 + (0.579 + 0.937i)T + (-0.447 + 0.894i)T^{2} \)
5 \( 1 + (0.999 - 0.0299i)T^{2} \)
7 \( 1 + (0.978 + 0.207i)T^{2} \)
13 \( 1 + (0.646 - 0.762i)T^{2} \)
17 \( 1 + (-1.02 - 1.45i)T + (-0.337 + 0.941i)T^{2} \)
19 \( 1 + (1.20 - 1.18i)T + (0.0149 - 0.999i)T^{2} \)
23 \( 1 + (0.733 - 0.680i)T^{2} \)
29 \( 1 + (0.447 + 0.894i)T^{2} \)
31 \( 1 + (0.842 - 0.538i)T^{2} \)
37 \( 1 + (-0.669 + 0.743i)T^{2} \)
41 \( 1 + (1.71 - 0.311i)T + (0.936 - 0.351i)T^{2} \)
47 \( 1 + (-0.858 - 0.512i)T^{2} \)
53 \( 1 + (-0.525 + 0.850i)T^{2} \)
59 \( 1 + (0.973 - 0.850i)T + (0.134 - 0.990i)T^{2} \)
61 \( 1 + (0.842 + 0.538i)T^{2} \)
67 \( 1 + (1.80 - 0.272i)T + (0.955 - 0.294i)T^{2} \)
71 \( 1 + (-0.992 - 0.119i)T^{2} \)
73 \( 1 + (-1.32 + 0.159i)T + (0.971 - 0.237i)T^{2} \)
79 \( 1 + (-0.913 + 0.406i)T^{2} \)
83 \( 1 + (1.34 - 0.327i)T + (0.887 - 0.460i)T^{2} \)
89 \( 1 + (-0.0628 + 0.838i)T + (-0.988 - 0.149i)T^{2} \)
97 \( 1 + (-1.01 - 1.54i)T + (-0.393 + 0.919i)T^{2} \)
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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

−8.634172423163826630226788760173, −8.053129879590210678823680189067, −7.47672347650106668429345880356, −6.65176514525497642296973131526, −6.18235656822435986221936769845, −5.64979992651712619077251836228, −4.54572840349990392619288091855, −3.64292165423878297884860947636, −1.75843617739605016072088967707, −1.55366744116345101963530728953, 0.30132369168494382303784312639, 1.78197103459129778098349971638, 2.98597681608971610327070423117, 3.66962354688752463684778912895, 4.59015295780863519629478720791, 5.17530532122421078954424723729, 6.24002712293421992728199129138, 7.06303787176798162100488010222, 7.908585068769146096537433524961, 8.667042564980019607894081077215

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