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Properties

Label 2-3784-3784.971-c0-0-0
Degree $2$
Conductor $3784$
Sign $0.950 + 0.311i$
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  + (0.983 − 0.178i)2-s + (0.0251 − 0.0861i)3-s + (0.936 − 0.351i)4-s + (0.00937 − 0.0892i)6-s + (0.858 − 0.512i)8-s + (0.835 + 0.534i)9-s + (−0.280 + 0.959i)11-s + (−0.00670 − 0.0894i)12-s + (0.753 − 0.657i)16-s + (−0.298 − 1.51i)17-s + (0.917 + 0.376i)18-s + (−0.327 − 0.0195i)19-s + (−0.104 + 0.994i)22-s + (−0.0225 − 0.0868i)24-s + (0.971 − 0.237i)25-s + ⋯
L(s)  = 1  + (0.983 − 0.178i)2-s + (0.0251 − 0.0861i)3-s + (0.936 − 0.351i)4-s + (0.00937 − 0.0892i)6-s + (0.858 − 0.512i)8-s + (0.835 + 0.534i)9-s + (−0.280 + 0.959i)11-s + (−0.00670 − 0.0894i)12-s + (0.753 − 0.657i)16-s + (−0.298 − 1.51i)17-s + (0.917 + 0.376i)18-s + (−0.327 − 0.0195i)19-s + (−0.104 + 0.994i)22-s + (−0.0225 − 0.0868i)24-s + (0.971 − 0.237i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3784\)    =    \(2^{3} \cdot 11 \cdot 43\)
Sign: $0.950 + 0.311i$
Analytic conductor: \(1.88846\)
Root analytic conductor: \(1.37421\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3784} (971, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3784,\ (\ :0),\ 0.950 + 0.311i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.645259152\)
\(L(\frac12)\) \(\approx\) \(2.645259152\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.983 + 0.178i)T \)
11 \( 1 + (0.280 - 0.959i)T \)
43 \( 1 + (0.999 + 0.0299i)T \)
good3 \( 1 + (-0.0251 + 0.0861i)T + (-0.842 - 0.538i)T^{2} \)
5 \( 1 + (-0.971 + 0.237i)T^{2} \)
7 \( 1 + (0.104 - 0.994i)T^{2} \)
13 \( 1 + (-0.791 + 0.611i)T^{2} \)
17 \( 1 + (0.298 + 1.51i)T + (-0.925 + 0.379i)T^{2} \)
19 \( 1 + (0.327 + 0.0195i)T + (0.992 + 0.119i)T^{2} \)
23 \( 1 + (-0.955 - 0.294i)T^{2} \)
29 \( 1 + (0.842 - 0.538i)T^{2} \)
31 \( 1 + (0.163 - 0.986i)T^{2} \)
37 \( 1 + (-0.913 + 0.406i)T^{2} \)
41 \( 1 + (0.161 - 1.18i)T + (-0.963 - 0.266i)T^{2} \)
47 \( 1 + (0.393 + 0.919i)T^{2} \)
53 \( 1 + (0.280 + 0.959i)T^{2} \)
59 \( 1 + (-1.35 - 0.811i)T + (0.473 + 0.880i)T^{2} \)
61 \( 1 + (0.163 + 0.986i)T^{2} \)
67 \( 1 + (0.714 - 1.82i)T + (-0.733 - 0.680i)T^{2} \)
71 \( 1 + (-0.575 - 0.817i)T^{2} \)
73 \( 1 + (-1.05 + 1.49i)T + (-0.337 - 0.941i)T^{2} \)
79 \( 1 + (0.978 - 0.207i)T^{2} \)
83 \( 1 + (0.663 + 1.85i)T + (-0.772 + 0.635i)T^{2} \)
89 \( 1 + (1.56 + 1.06i)T + (0.365 + 0.930i)T^{2} \)
97 \( 1 + (-0.0896 + 1.99i)T + (-0.995 - 0.0896i)T^{2} \)
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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

−8.561585673502973365947151247764, −7.52357641331534560023213341379, −7.13437331485918263877871048463, −6.49057225555149343379357187843, −5.42385842456512404855193723794, −4.66240199550569795551980050197, −4.38360352273938599010068098923, −3.06953725051533267223430075492, −2.37136342804725283004678192973, −1.38035969081026537265122053089, 1.39347674231523887711114757110, 2.47190771905492239481772831128, 3.63143802833067507398458054837, 3.90799961799958365179650722408, 5.00967003113504027097779547085, 5.64037787322869462318114334846, 6.66298947364531958124133770587, 6.77790526390652935290210913249, 8.050859837479595043105545512252, 8.426027267476441898940907760759

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