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Properties

Label 2-385-7.4-c1-0-1
Degree $2$
Conductor $385$
Sign $-0.160 - 0.987i$
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  + (−0.563 − 0.975i)2-s + (−0.761 + 1.31i)3-s + (0.365 − 0.632i)4-s + (−0.5 − 0.866i)5-s + 1.71·6-s + (−0.779 + 2.52i)7-s − 3.07·8-s + (0.340 + 0.589i)9-s + (−0.563 + 0.975i)10-s + (−0.5 + 0.866i)11-s + (0.556 + 0.963i)12-s − 6.04·13-s + (2.90 − 0.663i)14-s + 1.52·15-s + (1.00 + 1.73i)16-s + (0.556 − 0.963i)17-s + ⋯
L(s)  = 1  + (−0.398 − 0.689i)2-s + (−0.439 + 0.761i)3-s + (0.182 − 0.316i)4-s + (−0.223 − 0.387i)5-s + 0.700·6-s + (−0.294 + 0.955i)7-s − 1.08·8-s + (0.113 + 0.196i)9-s + (−0.178 + 0.308i)10-s + (−0.150 + 0.261i)11-s + (0.160 + 0.278i)12-s − 1.67·13-s + (0.776 − 0.177i)14-s + 0.393·15-s + (0.250 + 0.434i)16-s + (0.134 − 0.233i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.160 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.160 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $-0.160 - 0.987i$
Analytic conductor: \(3.07424\)
Root analytic conductor: \(1.75335\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 385,\ (\ :1/2),\ -0.160 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.288616 + 0.339321i\)
\(L(\frac12)\) \(\approx\) \(0.288616 + 0.339321i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.779 - 2.52i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.563 + 0.975i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.761 - 1.31i)T + (-1.5 - 2.59i)T^{2} \)
13 \( 1 + 6.04T + 13T^{2} \)
17 \( 1 + (-0.556 + 0.963i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.07 - 3.58i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.914 - 1.58i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.03T + 29T^{2} \)
31 \( 1 + (5.16 - 8.95i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.70 - 6.42i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.44T + 41T^{2} \)
43 \( 1 + 5.56T + 43T^{2} \)
47 \( 1 + (-2.25 - 3.90i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.29 + 9.17i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.72 + 6.44i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.22 + 5.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.83 + 11.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.84T + 71T^{2} \)
73 \( 1 + (5.23 - 9.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.750 + 1.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 + (7.02 + 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.85T + 97T^{2} \)
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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

−11.61489390489411979128710236204, −10.51700252169224808129324046823, −9.801897564233223710287174867020, −9.361259033555574406056629513052, −8.110346597559720599624642139261, −6.81214315158459713765032051981, −5.39394348525556823069581552464, −4.98349384621051016191989753045, −3.24706578532321264331961334544, −1.94177675224386588651547341635, 0.33188361232617896427490116059, 2.63850001056263620104679064602, 4.01102215612376295315734664996, 5.64144735596492677764104491755, 6.76264698784445853072692088691, 7.25764809837673494040617029420, 7.77986828932230953425830630112, 9.176617380642421883609251685402, 10.06899702400543611062764797682, 11.24733544453252949210128679894

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