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Properties

Label 2-385-385.54-c1-0-3
Degree $2$
Conductor $385$
Sign $-0.469 - 0.882i$
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  + (−1.15 + 2.00i)2-s + (−1.18 − 2.04i)3-s + (−1.67 − 2.89i)4-s + (−2.07 − 0.824i)5-s + 5.46·6-s + (−2.64 + 0.137i)7-s + 3.11·8-s + (−1.29 + 2.24i)9-s + (4.05 − 3.20i)10-s + (3.03 − 1.34i)11-s + (−3.95 + 6.85i)12-s + 0.917i·13-s + (2.77 − 5.44i)14-s + (0.768 + 5.22i)15-s + (−0.253 + 0.439i)16-s + (−0.346 + 0.200i)17-s + ⋯
L(s)  = 1  + (−0.817 + 1.41i)2-s + (−0.682 − 1.18i)3-s + (−0.836 − 1.44i)4-s + (−0.929 − 0.368i)5-s + 2.23·6-s + (−0.998 + 0.0519i)7-s + 1.10·8-s + (−0.431 + 0.747i)9-s + (1.28 − 1.01i)10-s + (0.913 − 0.406i)11-s + (−1.14 + 1.97i)12-s + 0.254i·13-s + (0.742 − 1.45i)14-s + (0.198 + 1.35i)15-s + (−0.0634 + 0.109i)16-s + (−0.0840 + 0.0485i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 - 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.469 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $-0.469 - 0.882i$
Analytic conductor: \(3.07424\)
Root analytic conductor: \(1.75335\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 385,\ (\ :1/2),\ -0.469 - 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.147999 + 0.246466i\)
\(L(\frac12)\) \(\approx\) \(0.147999 + 0.246466i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.07 + 0.824i)T \)
7 \( 1 + (2.64 - 0.137i)T \)
11 \( 1 + (-3.03 + 1.34i)T \)
good2 \( 1 + (1.15 - 2.00i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.18 + 2.04i)T + (-1.5 + 2.59i)T^{2} \)
13 \( 1 - 0.917iT - 13T^{2} \)
17 \( 1 + (0.346 - 0.200i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.02 - 5.24i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.41 - 3.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.75iT - 29T^{2} \)
31 \( 1 + (7.21 - 4.16i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.160 + 0.0927i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.12T + 41T^{2} \)
43 \( 1 + 0.604T + 43T^{2} \)
47 \( 1 + (-4.21 + 7.29i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.53 + 2.61i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.21 - 3.01i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.44 - 7.70i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.83 + 2.79i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.94T + 71T^{2} \)
73 \( 1 + (6.10 - 3.52i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.02 - 1.16i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.62iT - 83T^{2} \)
89 \( 1 + (-13.6 - 7.88i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.0T + 97T^{2} \)
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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

−11.82513626840510512142847552782, −10.68229102202032296598660081590, −9.265268377810650407901985137676, −8.718189872155906918286719852572, −7.57766981166190715977183182101, −7.01934638748428769218281946313, −6.28986165451983185515351267958, −5.47734589332478069261391340546, −3.73825768586407130832576191094, −1.07518359977726407862167474910, 0.34222010500279218588701785047, 2.75858258166646179838733948573, 3.81515949258580289356891893785, 4.51765476871522326235705303489, 6.25952583538829743470227101933, 7.45270876255381631213412891065, 9.028832568914369386991465401153, 9.313653743167597613235268355356, 10.40127652116894560967874433438, 10.92009467464941769057212832442

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