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Properties

Label 2-385-7.4-c1-0-11
Degree $2$
Conductor $385$
Sign $-0.761 + 0.648i$
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  + (1.36 + 2.36i)2-s + (−1.13 + 1.97i)3-s + (−2.71 + 4.70i)4-s + (0.5 + 0.866i)5-s − 6.21·6-s + (2.56 − 0.641i)7-s − 9.37·8-s + (−1.09 − 1.90i)9-s + (−1.36 + 2.36i)10-s + (0.5 − 0.866i)11-s + (−6.19 − 10.7i)12-s + 5.46·13-s + (5.01 + 5.18i)14-s − 2.27·15-s + (−7.34 − 12.7i)16-s + (2.36 − 4.09i)17-s + ⋯
L(s)  = 1  + (0.964 + 1.67i)2-s + (−0.658 + 1.13i)3-s + (−1.35 + 2.35i)4-s + (0.223 + 0.387i)5-s − 2.53·6-s + (0.970 − 0.242i)7-s − 3.31·8-s + (−0.366 − 0.634i)9-s + (−0.431 + 0.746i)10-s + (0.150 − 0.261i)11-s + (−1.78 − 3.09i)12-s + 1.51·13-s + (1.34 + 1.38i)14-s − 0.588·15-s + (−1.83 − 3.18i)16-s + (0.573 − 0.992i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $-0.761 + 0.648i$
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: \((2,\ 385,\ (\ :1/2),\ -0.761 + 0.648i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.646112 - 1.75595i\)
\(L(\frac12)\) \(\approx\) \(0.646112 - 1.75595i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.56 + 0.641i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-1.36 - 2.36i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.13 - 1.97i)T + (-1.5 - 2.59i)T^{2} \)
13 \( 1 - 5.46T + 13T^{2} \)
17 \( 1 + (-2.36 + 4.09i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.18 - 2.04i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.590 - 1.02i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.61T + 29T^{2} \)
31 \( 1 + (-3.47 + 6.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.70 + 4.68i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.79T + 41T^{2} \)
43 \( 1 + 9.46T + 43T^{2} \)
47 \( 1 + (2.80 + 4.85i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.38 + 4.13i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.615 - 1.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.57 - 9.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.03 - 10.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.0153T + 71T^{2} \)
73 \( 1 + (-0.0687 + 0.118i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.43 - 11.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + (1.80 + 3.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.86T + 97T^{2} \)
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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

−11.72806209471498760563499023169, −11.25612533495496553682709302636, −9.979431620129018581480543754812, −8.869449144290953772325782400998, −7.941224958117586587976698974150, −6.98718604932996552012701245908, −5.74853028305683000150726527824, −5.40026335575412680645330921875, −4.25365244854248137952801891324, −3.53987384249020759606279761532, 1.22173040919321771721800204926, 1.78877502745950537541675399424, 3.50009128200545019278621607528, 4.77384501527731850965301875730, 5.66278081433685429601333084602, 6.45626320645163300669891909623, 8.174026865640201272091751338576, 9.155948881956377133319455901367, 10.41805890417789845368443971577, 11.16888259719852431077149567072

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