Properties

Label 2-385-11.9-c1-0-22
Degree $2$
Conductor $385$
Sign $-0.957 + 0.288i$
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  + (0.690 − 2.12i)2-s + (1.30 − 0.951i)3-s + (−2.42 − 1.76i)4-s + (−0.309 − 0.951i)5-s + (−1.11 − 3.44i)6-s + (0.809 + 0.587i)7-s + (−1.80 + 1.31i)8-s + (−0.118 + 0.363i)9-s − 2.23·10-s + (−2.54 − 2.12i)11-s − 4.85·12-s + (0.736 − 2.26i)13-s + (1.80 − 1.31i)14-s + (−1.30 − 0.951i)15-s + (−0.309 − 0.951i)16-s + (1.04 + 3.21i)17-s + ⋯
L(s)  = 1  + (0.488 − 1.50i)2-s + (0.755 − 0.549i)3-s + (−1.21 − 0.881i)4-s + (−0.138 − 0.425i)5-s + (−0.456 − 1.40i)6-s + (0.305 + 0.222i)7-s + (−0.639 + 0.464i)8-s + (−0.0393 + 0.121i)9-s − 0.707·10-s + (−0.767 − 0.641i)11-s − 1.40·12-s + (0.204 − 0.628i)13-s + (0.483 − 0.351i)14-s + (−0.337 − 0.245i)15-s + (−0.0772 − 0.237i)16-s + (0.253 + 0.780i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $-0.957 + 0.288i$
Analytic conductor: \(3.07424\)
Root analytic conductor: \(1.75335\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 385,\ (\ :1/2),\ -0.957 + 0.288i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.298064 - 2.02578i\)
\(L(\frac12)\) \(\approx\) \(0.298064 - 2.02578i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (2.54 + 2.12i)T \)
good2 \( 1 + (-0.690 + 2.12i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-1.30 + 0.951i)T + (0.927 - 2.85i)T^{2} \)
13 \( 1 + (-0.736 + 2.26i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.04 - 3.21i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1 - 0.726i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 4.47T + 23T^{2} \)
29 \( 1 + (-4.73 - 3.44i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.381 - 1.17i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-2.23 - 1.62i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-5.23 + 3.80i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 0.763T + 43T^{2} \)
47 \( 1 + (-6.35 + 4.61i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.23 - 9.95i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-3.85 - 2.80i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.85 + 8.78i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 14.1T + 67T^{2} \)
71 \( 1 + (-0.208 - 0.640i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (6.97 + 5.06i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (4.26 - 13.1i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.26 + 10.0i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 17.2T + 89T^{2} \)
97 \( 1 + (-0.100 + 0.310i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.81871360819113563437085169509, −10.50548110551629010821743315251, −9.066074106837625225661995241798, −8.366881108472514915571235459804, −7.47435133098039371807705622699, −5.69451311749721211259173591532, −4.70839844352746932837507132591, −3.35631736301862010232394160492, −2.53459255338271896022783666608, −1.24784745694020690069363254571, 2.74775520243217931695804236082, 4.13496085109224146090348116090, 4.86112180311145715364746575055, 6.12917883158057356517935542873, 7.11475540424131652467184041204, 7.81487448181098685565717884188, 8.747886137935720709431429350992, 9.610641997105696863806906400518, 10.69587815479883717010490791087, 11.82617190786002253864772563105

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