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Properties

Label 2-816-204.191-c1-0-13
Degree $2$
Conductor $816$
Sign $0.175 - 0.984i$
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  + (1.61 + 0.618i)3-s + (1.73 + 1.73i)5-s + (−1.73 + 1.73i)7-s + (2.23 + 2.00i)9-s + (−1 − i)11-s − 2·13-s + (1.73 + 3.87i)15-s + (2.23 + 3.46i)17-s + 1.00·19-s + (−3.87 + 1.73i)21-s + (1.87 + 1.87i)23-s + 0.999i·25-s + (2.38 + 4.61i)27-s + (2.74 + 2.74i)29-s + (−6.20 − 6.20i)31-s + ⋯
L(s)  = 1  + (0.934 + 0.356i)3-s + (0.774 + 0.774i)5-s + (−0.654 + 0.654i)7-s + (0.745 + 0.666i)9-s + (−0.301 − 0.301i)11-s − 0.554·13-s + (0.447 + 1.00i)15-s + (0.542 + 0.840i)17-s + 0.231·19-s + (−0.845 + 0.377i)21-s + (0.390 + 0.390i)23-s + 0.199i·25-s + (0.458 + 0.888i)27-s + (0.508 + 0.508i)29-s + (−1.11 − 1.11i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(816\)    =    \(2^{4} \cdot 3 \cdot 17\)
Sign: $0.175 - 0.984i$
Analytic conductor: \(6.51579\)
Root analytic conductor: \(2.55260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{816} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 816,\ (\ :1/2),\ 0.175 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68131 + 1.40876i\)
\(L(\frac12)\) \(\approx\) \(1.68131 + 1.40876i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.61 - 0.618i)T \)
17 \( 1 + (-2.23 - 3.46i)T \)
good5 \( 1 + (-1.73 - 1.73i)T + 5iT^{2} \)
7 \( 1 + (1.73 - 1.73i)T - 7iT^{2} \)
11 \( 1 + (1 + i)T + 11iT^{2} \)
13 \( 1 + 2T + 13T^{2} \)
19 \( 1 - 1.00T + 19T^{2} \)
23 \( 1 + (-1.87 - 1.87i)T + 23iT^{2} \)
29 \( 1 + (-2.74 - 2.74i)T + 29iT^{2} \)
31 \( 1 + (6.20 + 6.20i)T + 31iT^{2} \)
37 \( 1 + (3.87 - 3.87i)T - 37iT^{2} \)
41 \( 1 + (-6.70 + 6.70i)T - 41iT^{2} \)
43 \( 1 + 1.00T + 43T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 - 11.4T + 53T^{2} \)
59 \( 1 + 11.7iT - 59T^{2} \)
61 \( 1 + (-0.127 - 0.127i)T + 61iT^{2} \)
67 \( 1 - 3.46iT - 67T^{2} \)
71 \( 1 + (5.87 - 5.87i)T - 71iT^{2} \)
73 \( 1 + (-10.7 + 10.7i)T - 73iT^{2} \)
79 \( 1 + (-6.20 + 6.20i)T - 79iT^{2} \)
83 \( 1 + 3.74iT - 83T^{2} \)
89 \( 1 - 2.01iT - 89T^{2} \)
97 \( 1 + (3 - 3i)T - 97iT^{2} \)
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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

−10.24335505863089431502063454008, −9.593722245228995838626995016064, −8.922393349886477878413093240144, −7.932914385392335764643651049775, −7.03501838760188838347025547139, −6.04410874009931855958890087879, −5.19037403406409845409312966375, −3.71693525073378734999962694808, −2.87915437511110632394474294997, −2.02778867462481174251895376001, 1.01503960706103860812881841894, 2.35216542936938217678741036295, 3.40376281291749667084468057531, 4.61803514002317973452108462649, 5.58502582034753413372988655133, 6.86961445629462900666441324536, 7.39494451473066915814643993639, 8.453752139899614369722791810380, 9.337204244157793708219556278956, 9.759614264034052713007997817829

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