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Properties

Label 2-385-385.384-c1-0-14
Degree $2$
Conductor $385$
Sign $0.161 - 0.986i$
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  + 0.826·2-s − 1.31·4-s + 2.23i·5-s + (2.61 + 0.428i)7-s − 2.74·8-s − 3·9-s + 1.84i·10-s + 3.31·11-s + 6.26i·13-s + (2.15 + 0.353i)14-s + 0.366·16-s + 4.55i·17-s − 2.47·18-s − 2.94i·20-s + 2.74·22-s + ⋯
L(s)  = 1  + 0.584·2-s − 0.658·4-s + 0.999i·5-s + (0.986 + 0.161i)7-s − 0.969·8-s − 9-s + 0.584i·10-s + 1.00·11-s + 1.73i·13-s + (0.576 + 0.0946i)14-s + 0.0916·16-s + 1.10i·17-s − 0.584·18-s − 0.658i·20-s + 0.584·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $0.161 - 0.986i$
Analytic conductor: \(3.07424\)
Root analytic conductor: \(1.75335\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (384, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 385,\ (\ :1/2),\ 0.161 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10752 + 0.940673i\)
\(L(\frac12)\) \(\approx\) \(1.10752 + 0.940673i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 2.23iT \)
7 \( 1 + (-2.61 - 0.428i)T \)
11 \( 1 - 3.31T \)
good2 \( 1 - 0.826T + 2T^{2} \)
3 \( 1 + 3T^{2} \)
13 \( 1 - 6.26iT - 13T^{2} \)
17 \( 1 - 4.55iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 8.94iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 8.52T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 14.8iT - 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 17.0iT - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 13.3iT - 83T^{2} \)
89 \( 1 - 13.4iT - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−11.54518651620378752512436895465, −10.98329645833499499108145931622, −9.549577691334329539064354885758, −8.853648399130108548121570986286, −7.88127477594682902672107995306, −6.49817457793084734610831761809, −5.82435168169911861671642757543, −4.44991568866133644131740766719, −3.68365359391781900594526810719, −2.12205056179652581842040269851, 0.873782004627118070869113036577, 3.05323989650089221725162564962, 4.34505518242219243297883278280, 5.22424022276230332185509850148, 5.80661276715780154502083058168, 7.56157527293219250059935804687, 8.611511411796878729320056934272, 8.951639737515488579859664651302, 10.20422032464952422336087349501, 11.44587241051285564636191233271

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