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Properties

Label 2-418-209.65-c1-0-18
Degree $2$
Conductor $418$
Sign $-0.998 + 0.0470i$
Analytic cond. $3.33774$
Root an. cond. $1.82695$
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 + (−2.05 − 1.18i)3-s + (−0.499 − 0.866i)4-s + (2.05 − 3.56i)5-s + (−2.05 + 1.18i)6-s − 2.89i·7-s − 0.999·8-s + (1.32 + 2.29i)9-s + (−2.05 − 3.56i)10-s + (2.64 + 2.00i)11-s + 2.37i·12-s + (−0.104 − 0.180i)13-s + (−2.50 − 1.44i)14-s + (−8.47 + 4.89i)15-s + (−0.5 + 0.866i)16-s + (6.81 + 3.93i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−1.18 − 0.686i)3-s + (−0.249 − 0.433i)4-s + (0.920 − 1.59i)5-s + (−0.840 + 0.485i)6-s − 1.09i·7-s − 0.353·8-s + (0.441 + 0.765i)9-s + (−0.650 − 1.12i)10-s + (0.796 + 0.604i)11-s + 0.686i·12-s + (−0.0289 − 0.0501i)13-s + (−0.669 − 0.386i)14-s + (−2.18 + 1.26i)15-s + (−0.125 + 0.216i)16-s + (1.65 + 0.953i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0470i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-0.998 + 0.0470i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ -0.998 + 0.0470i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0294311 - 1.24917i\)
\(L(\frac12)\) \(\approx\) \(0.0294311 - 1.24917i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-2.64 - 2.00i)T \)
19 \( 1 + (4.34 - 0.327i)T \)
good3 \( 1 + (2.05 + 1.18i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.05 + 3.56i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 2.89iT - 7T^{2} \)
13 \( 1 + (0.104 + 0.180i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-6.81 - 3.93i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.98 + 5.17i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.71 - 2.96i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.73iT - 31T^{2} \)
37 \( 1 + 1.38iT - 37T^{2} \)
41 \( 1 + (0.769 - 1.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-8.75 - 5.05i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.26 - 2.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.55 + 2.63i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (11.8 + 6.85i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.468 - 0.270i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.14 + 0.662i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8.85 - 5.10i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.652 + 0.376i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.12 + 8.88i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.79iT - 83T^{2} \)
89 \( 1 + (1.51 - 0.875i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.47 - 3.74i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.71474695447153944479746367163, −10.16483866017588419152342500707, −9.150659393440304503529700101922, −8.010583659096489332165618459501, −6.59861951501755160207569912436, −5.88846943111105470772083124592, −4.92031270813782542622589996124, −4.06936318282639856419327761945, −1.64919685698595037151277775781, −0.918285290808364165494835219877, 2.57902124136696890153849494153, 3.84235170975209497724633730190, 5.46382361855105701881397755725, 5.84833068907285797968529510685, 6.50390791000099100221110153308, 7.66851619104184191015387233053, 9.217685444357459133794834310444, 9.901280661982875131217261554202, 10.79826497985499276312350412810, 11.64683323792405635393281217472

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