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Properties

Label 2-416-52.3-c0-0-0
Degree $2$
Conductor $416$
Sign $0.494 - 0.869i$
Analytic cond. $0.207611$
Root an. cond. $0.455643$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)3-s + (0.866 + 0.5i)7-s + (−0.866 + 0.5i)11-s + 13-s + (−0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s − 0.999·21-s + (0.866 − 0.5i)23-s − 25-s − i·27-s + (0.5 + 0.866i)29-s + (0.499 − 0.866i)33-s + (−0.5 − 0.866i)37-s + (−0.866 + 0.5i)39-s + (−0.5 − 0.866i)41-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (0.866 + 0.5i)7-s + (−0.866 + 0.5i)11-s + 13-s + (−0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s − 0.999·21-s + (0.866 − 0.5i)23-s − 25-s − i·27-s + (0.5 + 0.866i)29-s + (0.499 − 0.866i)33-s + (−0.5 − 0.866i)37-s + (−0.866 + 0.5i)39-s + (−0.5 − 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.494 - 0.869i$
Analytic conductor: \(0.207611\)
Root analytic conductor: \(0.455643\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :0),\ 0.494 - 0.869i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6760501287\)
\(L(\frac12)\) \(\approx\) \(0.6760501287\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 - T \)
good3 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + 2iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 2iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−11.45407323113429728214257958095, −10.71453289582306159245829494502, −10.12554656228783781527044351141, −8.726316963686132906655453740779, −8.082139972157866462866888207237, −6.78913577969456354190091977661, −5.50067776687963736586586474262, −5.12924886940871015484160370661, −3.80493148432603117085106764238, −2.01481993602685396123698186653, 1.16817482024152375815029543823, 3.09689828821238581465743595102, 4.67881471752009105687629546664, 5.55519337493800148989496338698, 6.53658872710570387340612681246, 7.52366621259775273015933246989, 8.362350221878164128253945514060, 9.525949676586403847380620967776, 10.71647469203674048902874289702, 11.46235892079518807258944203133

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