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Properties

Label 2-385-11.3-c1-0-0
Degree $2$
Conductor $385$
Sign $0.153 + 0.988i$
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  + (−2.14 + 1.55i)2-s + (0.669 + 2.06i)3-s + (1.54 − 4.75i)4-s + (−0.809 − 0.587i)5-s + (−4.63 − 3.36i)6-s + (−0.309 + 0.951i)7-s + (2.45 + 7.56i)8-s + (−1.36 + 0.995i)9-s + 2.64·10-s + (−3.28 − 0.449i)11-s + 10.8·12-s + (−2.52 + 1.83i)13-s + (−0.817 − 2.51i)14-s + (0.669 − 2.06i)15-s + (−8.92 − 6.48i)16-s + (−4.54 − 3.30i)17-s + ⋯
L(s)  = 1  + (−1.51 + 1.09i)2-s + (0.386 + 1.18i)3-s + (0.773 − 2.37i)4-s + (−0.361 − 0.262i)5-s + (−1.89 − 1.37i)6-s + (−0.116 + 0.359i)7-s + (0.868 + 2.67i)8-s + (−0.456 + 0.331i)9-s + 0.836·10-s + (−0.990 − 0.135i)11-s + 3.12·12-s + (−0.701 + 0.509i)13-s + (−0.218 − 0.672i)14-s + (0.172 − 0.531i)15-s + (−2.23 − 1.62i)16-s + (−1.10 − 0.801i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.153 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $0.153 + 0.988i$
Analytic conductor: \(3.07424\)
Root analytic conductor: \(1.75335\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 385,\ (\ :1/2),\ 0.153 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0513849 - 0.0439979i\)
\(L(\frac12)\) \(\approx\) \(0.0513849 - 0.0439979i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.809 + 0.587i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (3.28 + 0.449i)T \)
good2 \( 1 + (2.14 - 1.55i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.669 - 2.06i)T + (-2.42 + 1.76i)T^{2} \)
13 \( 1 + (2.52 - 1.83i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (4.54 + 3.30i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.02 + 6.22i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 5.20T + 23T^{2} \)
29 \( 1 + (2.59 - 7.99i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.937 + 0.680i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.92 + 8.99i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.01 - 6.21i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 5.24T + 43T^{2} \)
47 \( 1 + (-0.813 - 2.50i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.923 + 0.670i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.874 + 2.69i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-7.04 - 5.11i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 2.54T + 67T^{2} \)
71 \( 1 + (9.19 + 6.68i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.74 - 8.45i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (2.64 - 1.92i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-0.583 - 0.424i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 0.247T + 89T^{2} \)
97 \( 1 + (5.95 - 4.32i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.43020542829743240719346961183, −10.70747951970035222271322157087, −9.828348435655187220469499924070, −9.116863311935556053933402823319, −8.672040587370417665734739290793, −7.56026518312362685828790463535, −6.71815362967276985782734597988, −5.33607145065450187053063511633, −4.52254681976558322520139841330, −2.49357344789893835232914725852, 0.06490002371088956051537061621, 1.82387605409428555176816292215, 2.63184009108098434571440883324, 4.01191092335393951791082236285, 6.36368562820258439890876187706, 7.49439913657957258321354704850, 7.945057291271810368570423981596, 8.547395513254901443698952582858, 10.11951001819001878280928309131, 10.27347982872915782723376003944

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