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Properties

Label 2-385-11.3-c1-0-21
Degree $2$
Conductor $385$
Sign $-0.794 + 0.607i$
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  + (1.76 − 1.28i)2-s + (−0.720 − 2.21i)3-s + (0.850 − 2.61i)4-s + (0.809 + 0.587i)5-s + (−4.11 − 2.98i)6-s + (0.309 − 0.951i)7-s + (−0.507 − 1.56i)8-s + (−1.97 + 1.43i)9-s + 2.18·10-s + (−1.17 − 3.10i)11-s − 6.42·12-s + (−1.57 + 1.14i)13-s + (−0.673 − 2.07i)14-s + (0.720 − 2.21i)15-s + (1.55 + 1.13i)16-s + (−2.34 − 1.70i)17-s + ⋯
L(s)  = 1  + (1.24 − 0.906i)2-s + (−0.416 − 1.28i)3-s + (0.425 − 1.30i)4-s + (0.361 + 0.262i)5-s + (−1.67 − 1.21i)6-s + (0.116 − 0.359i)7-s + (−0.179 − 0.552i)8-s + (−0.657 + 0.477i)9-s + 0.689·10-s + (−0.354 − 0.935i)11-s − 1.85·12-s + (−0.437 + 0.318i)13-s + (−0.180 − 0.554i)14-s + (0.186 − 0.572i)15-s + (0.389 + 0.282i)16-s + (−0.567 − 0.412i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.794 + 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $-0.794 + 0.607i$
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: \((2,\ 385,\ (\ :1/2),\ -0.794 + 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.754451 - 2.22991i\)
\(L(\frac12)\) \(\approx\) \(0.754451 - 2.22991i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.17 + 3.10i)T \)
good2 \( 1 + (-1.76 + 1.28i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.720 + 2.21i)T + (-2.42 + 1.76i)T^{2} \)
13 \( 1 + (1.57 - 1.14i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.34 + 1.70i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.21 - 6.82i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 7.16T + 23T^{2} \)
29 \( 1 + (1.51 - 4.66i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (3.03 - 2.20i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.38 + 7.35i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.39 + 4.28i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 10.2T + 43T^{2} \)
47 \( 1 + (-0.664 - 2.04i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (8.17 - 5.94i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.00 + 6.18i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.44 + 2.50i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 5.75T + 67T^{2} \)
71 \( 1 + (-0.854 - 0.620i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.488 + 1.50i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (9.89 - 7.18i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-7.49 - 5.44i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 12.2T + 89T^{2} \)
97 \( 1 + (3.52 - 2.56i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.05891511853919876161346109598, −10.80472955713896531406628423965, −9.324535026618964259928120743987, −7.86508024687004845297051097869, −6.95642470294108703918409139320, −5.91879596126436035481456099382, −5.17695108045112876314558268118, −3.70396319633766454982042631366, −2.51054051334740310173323681085, −1.29902249591068879228768215086, 2.82866573690763899973621434338, 4.31991187440442300551003490954, 4.88967929319151433235730174236, 5.50846283879324851082855918036, 6.66377334115827182359809481040, 7.65647723792464216943965887714, 9.130689606438261938922671021021, 9.806119636316846274567665494906, 10.86150986809319586076742600503, 11.77637891377184755103841030376

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