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Properties

Label 2-816-12.11-c1-0-2
Degree $2$
Conductor $816$
Sign $-0.989 + 0.146i$
Analytic cond. $6.51579$
Root an. cond. $2.55260$
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  + (0.254 + 1.71i)3-s + 2i·5-s − 0.508i·7-s + (−2.87 + 0.870i)9-s − 4.44·11-s − 1.74·13-s + (−3.42 + 0.508i)15-s + i·17-s − 1.01i·19-s + (0.870 − 0.129i)21-s − 6.34·23-s + 25-s + (−2.22 − 4.69i)27-s + 9.48i·29-s − 4.44i·31-s + ⋯
L(s)  = 1  + (0.146 + 0.989i)3-s + 0.894i·5-s − 0.192i·7-s + (−0.956 + 0.290i)9-s − 1.33·11-s − 0.483·13-s + (−0.884 + 0.131i)15-s + 0.242i·17-s − 0.233i·19-s + (0.190 − 0.0281i)21-s − 1.32·23-s + 0.200·25-s + (−0.427 − 0.903i)27-s + 1.76i·29-s − 0.798i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.146i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(816\)    =    \(2^{4} \cdot 3 \cdot 17\)
Sign: $-0.989 + 0.146i$
Analytic conductor: \(6.51579\)
Root analytic conductor: \(2.55260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{816} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 816,\ (\ :1/2),\ -0.989 + 0.146i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0546514 - 0.740913i\)
\(L(\frac12)\) \(\approx\) \(0.0546514 - 0.740913i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.254 - 1.71i)T \)
17 \( 1 - iT \)
good5 \( 1 - 2iT - 5T^{2} \)
7 \( 1 + 0.508iT - 7T^{2} \)
11 \( 1 + 4.44T + 11T^{2} \)
13 \( 1 + 1.74T + 13T^{2} \)
19 \( 1 + 1.01iT - 19T^{2} \)
23 \( 1 + 6.34T + 23T^{2} \)
29 \( 1 - 9.48iT - 29T^{2} \)
31 \( 1 + 4.44iT - 31T^{2} \)
37 \( 1 - 5.48T + 37T^{2} \)
41 \( 1 + 5.48iT - 41T^{2} \)
43 \( 1 - 8.88iT - 43T^{2} \)
47 \( 1 + 5.83T + 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 + 1.01T + 59T^{2} \)
61 \( 1 + 9.48T + 61T^{2} \)
67 \( 1 - 2.03iT - 67T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 - 9.26iT - 79T^{2} \)
83 \( 1 - 14.7T + 83T^{2} \)
89 \( 1 - 9.22iT - 89T^{2} \)
97 \( 1 - 5.48T + 97T^{2} \)
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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

−10.58602552211744027746055000456, −10.04889683455019629013641729221, −9.153091541637911990790713369463, −8.089984451400819442050733984447, −7.40990063899574920536988880908, −6.22225836484393019063155255464, −5.29033885100207806540108034463, −4.35767692772189586075244982287, −3.21832860117570209799117799553, −2.43014767362295847757848378921, 0.33299678212769762720148168354, 1.91408749048543060522335379025, 2.91482721386110105675223266527, 4.49090740942867606002061258439, 5.46235691516356393334928406408, 6.21915915037555927213112499805, 7.48092201936566715633547066580, 8.006065523431502773081282562769, 8.742894008822563586572764614284, 9.714684145200672841122911933138

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