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Properties

Label 2-618-103.46-c1-0-13
Degree $2$
Conductor $618$
Sign $-0.398 + 0.917i$
Analytic cond. $4.93475$
Root an. cond. $2.22143$
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  + (−0.5 − 0.866i)2-s − 3-s + (−0.499 + 0.866i)4-s + (1.93 − 3.35i)5-s + (0.5 + 0.866i)6-s + (−2.50 + 4.34i)7-s + 0.999·8-s + 9-s − 3.87·10-s + (2.13 − 3.70i)11-s + (0.499 − 0.866i)12-s + 2.62·13-s + 5.01·14-s + (−1.93 + 3.35i)15-s + (−0.5 − 0.866i)16-s + (2.26 − 3.92i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s − 0.577·3-s + (−0.249 + 0.433i)4-s + (0.866 − 1.50i)5-s + (0.204 + 0.353i)6-s + (−0.948 + 1.64i)7-s + 0.353·8-s + 0.333·9-s − 1.22·10-s + (0.644 − 1.11i)11-s + (0.144 − 0.249i)12-s + 0.726·13-s + 1.34·14-s + (−0.500 + 0.866i)15-s + (−0.125 − 0.216i)16-s + (0.549 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.398 + 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.398 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(618\)    =    \(2 \cdot 3 \cdot 103\)
Sign: $-0.398 + 0.917i$
Analytic conductor: \(4.93475\)
Root analytic conductor: \(2.22143\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{618} (355, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 618,\ (\ :1/2),\ -0.398 + 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.548716 - 0.836971i\)
\(L(\frac12)\) \(\approx\) \(0.548716 - 0.836971i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + T \)
103 \( 1 + (-9.32 + 4.00i)T \)
good5 \( 1 + (-1.93 + 3.35i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.50 - 4.34i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.13 + 3.70i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.62T + 13T^{2} \)
17 \( 1 + (-2.26 + 3.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.388 - 0.672i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.05T + 23T^{2} \)
29 \( 1 + (2.31 + 4.00i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10.1T + 31T^{2} \)
37 \( 1 + 0.866T + 37T^{2} \)
41 \( 1 + (3.89 + 6.74i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.11 + 3.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.38 + 2.39i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.59 - 9.69i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.64 + 11.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 7.15T + 61T^{2} \)
67 \( 1 + (0.294 - 0.510i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.74 + 4.75i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 10.6T + 73T^{2} \)
79 \( 1 + 6.56T + 79T^{2} \)
83 \( 1 + (0.764 + 1.32i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.42T + 89T^{2} \)
97 \( 1 + (-0.903 - 1.56i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.06396466825043120989822963787, −9.477249970452341599654795755062, −8.836944143721037594219375252316, −8.232468251320133822814184053721, −6.26384347640420238601495662844, −5.84923052029876482131466687896, −4.95790033169992622107086379877, −3.49086825803176408580309760165, −2.11030980430975919826774229053, −0.72368309907213651007350973406, 1.48852909528385556535426132126, 3.35802646825722771090328564812, 4.35290024024934329497983786992, 5.97612042785213713169169982484, 6.58465962847871492537932799481, 6.96338200756916256800100853258, 7.947096477816175143170127570476, 9.635939093276563845881031078280, 10.07008560571304903276620129736, 10.48935086671220361622614498332

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