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Properties

Label 2-618-103.76-c1-0-2
Degree $2$
Conductor $618$
Sign $-0.999 + 0.0212i$
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  + (0.0922 + 0.995i)2-s + (0.739 + 0.673i)3-s + (−0.982 + 0.183i)4-s + (−0.987 + 1.98i)5-s + (−0.602 + 0.798i)6-s + (−0.265 + 0.933i)7-s + (−0.273 − 0.961i)8-s + (0.0922 + 0.995i)9-s + (−2.06 − 0.800i)10-s + (−0.0650 − 0.701i)11-s + (−0.850 − 0.526i)12-s + (−0.743 + 2.61i)13-s + (−0.953 − 0.178i)14-s + (−2.06 + 0.800i)15-s + (0.932 − 0.361i)16-s + (−0.557 − 0.738i)17-s + ⋯
L(s)  = 1  + (0.0652 + 0.704i)2-s + (0.426 + 0.388i)3-s + (−0.491 + 0.0918i)4-s + (−0.441 + 0.886i)5-s + (−0.246 + 0.325i)6-s + (−0.100 + 0.352i)7-s + (−0.0967 − 0.340i)8-s + (0.0307 + 0.331i)9-s + (−0.653 − 0.253i)10-s + (−0.0196 − 0.211i)11-s + (−0.245 − 0.151i)12-s + (−0.206 + 0.724i)13-s + (−0.254 − 0.0476i)14-s + (−0.533 + 0.206i)15-s + (0.233 − 0.0903i)16-s + (−0.135 − 0.179i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0212i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(618\)    =    \(2 \cdot 3 \cdot 103\)
Sign: $-0.999 + 0.0212i$
Analytic conductor: \(4.93475\)
Root analytic conductor: \(2.22143\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{618} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 618,\ (\ :1/2),\ -0.999 + 0.0212i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0121103 - 1.13944i\)
\(L(\frac12)\) \(\approx\) \(0.0121103 - 1.13944i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0922 - 0.995i)T \)
3 \( 1 + (-0.739 - 0.673i)T \)
103 \( 1 + (-10.1 + 0.641i)T \)
good5 \( 1 + (0.987 - 1.98i)T + (-3.01 - 3.99i)T^{2} \)
7 \( 1 + (0.265 - 0.933i)T + (-5.95 - 3.68i)T^{2} \)
11 \( 1 + (0.0650 + 0.701i)T + (-10.8 + 2.02i)T^{2} \)
13 \( 1 + (0.743 - 2.61i)T + (-11.0 - 6.84i)T^{2} \)
17 \( 1 + (0.557 + 0.738i)T + (-4.65 + 16.3i)T^{2} \)
19 \( 1 + (5.09 - 4.64i)T + (1.75 - 18.9i)T^{2} \)
23 \( 1 + (-0.433 + 4.67i)T + (-22.6 - 4.22i)T^{2} \)
29 \( 1 + (-0.415 + 0.834i)T + (-17.4 - 23.1i)T^{2} \)
31 \( 1 + (3.13 - 1.21i)T + (22.9 - 20.8i)T^{2} \)
37 \( 1 + (1.87 + 1.16i)T + (16.4 + 33.1i)T^{2} \)
41 \( 1 + (1.71 + 3.44i)T + (-24.7 + 32.7i)T^{2} \)
43 \( 1 + (3.61 - 2.23i)T + (19.1 - 38.4i)T^{2} \)
47 \( 1 - 8.41T + 47T^{2} \)
53 \( 1 + (1.64 - 1.49i)T + (4.89 - 52.7i)T^{2} \)
59 \( 1 + (-2.84 - 10.0i)T + (-50.1 + 31.0i)T^{2} \)
61 \( 1 + (-2.50 - 3.32i)T + (-16.6 + 58.6i)T^{2} \)
67 \( 1 + (-0.463 - 1.62i)T + (-56.9 + 35.2i)T^{2} \)
71 \( 1 + (1.60 + 3.21i)T + (-42.7 + 56.6i)T^{2} \)
73 \( 1 + (-5.52 - 11.0i)T + (-43.9 + 58.2i)T^{2} \)
79 \( 1 + (-1.38 + 2.77i)T + (-47.6 - 63.0i)T^{2} \)
83 \( 1 + (-1.00 + 3.54i)T + (-70.5 - 43.6i)T^{2} \)
89 \( 1 + (-8.31 - 1.55i)T + (82.9 + 32.1i)T^{2} \)
97 \( 1 + (2.65 - 3.51i)T + (-26.5 - 93.2i)T^{2} \)
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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

−10.83380763531478133582265446621, −10.21634621533898807114372864656, −9.067750570320359323954982542966, −8.481404356923466872063320364343, −7.44971669840854078455693931661, −6.69439929634875682009910805108, −5.75466699813748572033081621793, −4.46504664918499319724022288975, −3.61716643019434631940727789194, −2.37859586332849845490586193828, 0.57523206657345482852662106896, 2.05324326827400681045189093696, 3.37767019044505141596464761617, 4.39198103984607884460296877541, 5.31082533043592532982576930083, 6.71328879666802354949614022603, 7.75514950995861797292519729179, 8.553605208662203183553324847239, 9.236049148113179119646588361510, 10.23346130449810467845748758593

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