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Properties

Label 2-306-17.16-c1-0-3
Degree $2$
Conductor $306$
Sign $0.727 - 0.685i$
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 + 2.82i·5-s + 8-s + 2.82i·10-s + 2.82i·11-s + 2·13-s + 16-s + (3 − 2.82i)17-s − 4·19-s + 2.82i·20-s + 2.82i·22-s − 5.65i·23-s − 3.00·25-s + 2·26-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.26i·5-s + 0.353·8-s + 0.894i·10-s + 0.852i·11-s + 0.554·13-s + 0.250·16-s + (0.727 − 0.685i)17-s − 0.917·19-s + 0.632i·20-s + 0.603i·22-s − 1.17i·23-s − 0.600·25-s + 0.392·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 306 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 306 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(306\)    =    \(2 \cdot 3^{2} \cdot 17\)
Sign: $0.727 - 0.685i$
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: \((2,\ 306,\ (\ :1/2),\ 0.727 - 0.685i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81836 + 0.722031i\)
\(L(\frac12)\) \(\approx\) \(1.81836 + 0.722031i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3 + 2.82i)T \)
good5 \( 1 - 2.82iT - 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 2.82iT - 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 5.65iT - 23T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 8.48iT - 37T^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 8.48iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 5.65iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 16.9iT - 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 16.9iT - 97T^{2} \)
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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

−11.93607501792769182668295031318, −10.75233353396644114749424326211, −10.42142763501684524597925503205, −9.053198130282691938059944934421, −7.62751826910890796606047095301, −6.85962790420539338101188665433, −5.98640210181096258611109775531, −4.62963716484723136444985201560, −3.41834532574028591770644406293, −2.26836199110576836875905430089, 1.40822723952589668292103229810, 3.35726652086495229374274367534, 4.48036019142334308243976671589, 5.53677329077078390105550412716, 6.35532558205704067565204749071, 7.940679935227730533037020557938, 8.586143756780708080869323228271, 9.720323945926297846589606220523, 10.89693382820758112729650844712, 11.77278203391953279524091178602

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