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Properties

Label 2-306-17.16-c1-0-5
Degree $2$
Conductor $306$
Sign $-0.242 + 0.970i$
Analytic cond. $2.44342$
Root an. cond. $1.56314$
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  − 2-s + 4-s − 2i·5-s − 2i·7-s − 8-s + 2i·10-s − 6·13-s + 2i·14-s + 16-s + (1 − 4i)17-s − 2i·20-s − 6i·23-s + 25-s + 6·26-s − 2i·28-s + 6i·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.894i·5-s − 0.755i·7-s − 0.353·8-s + 0.632i·10-s − 1.66·13-s + 0.534i·14-s + 0.250·16-s + (0.242 − 0.970i)17-s − 0.447i·20-s − 1.25i·23-s + 0.200·25-s + 1.17·26-s − 0.377i·28-s + 1.11i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 306 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 306 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(306\)    =    \(2 \cdot 3^{2} \cdot 17\)
Sign: $-0.242 + 0.970i$
Analytic conductor: \(2.44342\)
Root analytic conductor: \(1.56314\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{306} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 306,\ (\ :1/2),\ -0.242 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.462132 - 0.591888i\)
\(L(\frac12)\) \(\approx\) \(0.462132 - 0.591888i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
17 \( 1 + (-1 + 4i)T \)
good5 \( 1 + 2iT - 5T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 6T + 13T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 + 10iT - 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 10iT - 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 - 10iT - 71T^{2} \)
73 \( 1 - 16iT - 73T^{2} \)
79 \( 1 + 6iT - 79T^{2} \)
83 \( 1 - 16T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 12iT - 97T^{2} \)
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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

−11.39229639918401649114190043432, −10.27286135407401634577015460302, −9.583801770941831709224939310821, −8.679494410516895468290706512401, −7.60759724560153453618689919832, −6.90815654462119031721215776388, −5.33859710224337001497233358466, −4.34740552430899691417646475121, −2.53052775021901908505245268631, −0.67366562333104686285538858619, 2.11949469869452124569131126612, 3.27168808799425076794931789941, 5.11558943299602008089832484465, 6.30999695477665657506918913102, 7.25925873537568089145198026813, 8.136030851804176900264421522100, 9.296664444946094720234736753238, 10.05748357608719393733177976700, 10.88799211141688944599832329502, 11.91453850666806897406092960330

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