Properties

Label 2-385-7.2-c1-0-8
Degree $2$
Conductor $385$
Sign $0.792 - 0.609i$
Analytic cond. $3.07424$
Root an. cond. $1.75335$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.139 + 0.241i)2-s + (−0.137 − 0.238i)3-s + (0.961 + 1.66i)4-s + (−0.5 + 0.866i)5-s + 0.0770·6-s + (1.16 − 2.37i)7-s − 1.09·8-s + (1.46 − 2.53i)9-s + (−0.139 − 0.241i)10-s + (0.5 + 0.866i)11-s + (0.265 − 0.459i)12-s + 6.03·13-s + (0.412 + 0.612i)14-s + 0.275·15-s + (−1.76 + 3.06i)16-s + (2.24 + 3.88i)17-s + ⋯
L(s)  = 1  + (−0.0987 + 0.170i)2-s + (−0.0796 − 0.137i)3-s + (0.480 + 0.832i)4-s + (−0.223 + 0.387i)5-s + 0.0314·6-s + (0.439 − 0.898i)7-s − 0.387·8-s + (0.487 − 0.844i)9-s + (−0.0441 − 0.0764i)10-s + (0.150 + 0.261i)11-s + (0.0765 − 0.132i)12-s + 1.67·13-s + (0.110 + 0.163i)14-s + 0.0712·15-s + (−0.442 + 0.766i)16-s + (0.543 + 0.941i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 - 0.609i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.792 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $0.792 - 0.609i$
Analytic conductor: \(3.07424\)
Root analytic conductor: \(1.75335\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 385,\ (\ :1/2),\ 0.792 - 0.609i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41760 + 0.482130i\)
\(L(\frac12)\) \(\approx\) \(1.41760 + 0.482130i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-1.16 + 2.37i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.139 - 0.241i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.137 + 0.238i)T + (-1.5 + 2.59i)T^{2} \)
13 \( 1 - 6.03T + 13T^{2} \)
17 \( 1 + (-2.24 - 3.88i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.08 - 5.35i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.0705 + 0.122i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.06T + 29T^{2} \)
31 \( 1 + (-1.67 - 2.90i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.74 + 8.22i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.23T + 41T^{2} \)
43 \( 1 - 5.53T + 43T^{2} \)
47 \( 1 + (3.46 - 6.00i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.40 + 9.35i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.01 + 8.68i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.58 + 7.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.24 + 9.07i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.66T + 71T^{2} \)
73 \( 1 + (-2.03 - 3.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.24 - 12.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.71T + 83T^{2} \)
89 \( 1 + (8.92 - 15.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.310T + 97T^{2} \)
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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

−11.20361561503216592900357383804, −10.84772576275235091229851226943, −9.610677614735033102346054168933, −8.311727321785553745259051515168, −7.77561995420668665934667028950, −6.70580095794136222998326316365, −6.07379954124894128081205689914, −3.92777157750081047506829163431, −3.67233782428770207638646695170, −1.58687035403697862599843881822, 1.31747762092153205545378911171, 2.68108111914972807188604312101, 4.46066922556468615708223396225, 5.44597395763790719087476671008, 6.26597716338590454141059393896, 7.55173639017462066694102040970, 8.658474860244444652996680198731, 9.361729777463869657753776878608, 10.51232159049915721392564275390, 11.29348138506864815887983037504

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