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Properties

Label 2-816-17.4-c1-0-7
Degree $2$
Conductor $816$
Sign $0.273 + 0.961i$
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  + (−0.707 + 0.707i)3-s + (−2.87 + 2.87i)5-s + (−0.652 − 0.652i)7-s − 1.00i·9-s + (−1.60 − 1.60i)11-s − 3.57·13-s − 4.06i·15-s + (3.79 − 1.60i)17-s − 1.72i·19-s + 0.922·21-s + (3.60 + 3.60i)23-s − 11.5i·25-s + (0.707 + 0.707i)27-s + (−1.55 + 1.55i)29-s + (2.92 − 2.92i)31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−1.28 + 1.28i)5-s + (−0.246 − 0.246i)7-s − 0.333i·9-s + (−0.483 − 0.483i)11-s − 0.991·13-s − 1.04i·15-s + (0.921 − 0.389i)17-s − 0.396i·19-s + 0.201·21-s + (0.751 + 0.751i)23-s − 2.30i·25-s + (0.136 + 0.136i)27-s + (−0.289 + 0.289i)29-s + (0.524 − 0.524i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.273 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.273 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(816\)    =    \(2^{4} \cdot 3 \cdot 17\)
Sign: $0.273 + 0.961i$
Analytic conductor: \(6.51579\)
Root analytic conductor: \(2.55260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{816} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 816,\ (\ :1/2),\ 0.273 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.329153 - 0.248509i\)
\(L(\frac12)\) \(\approx\) \(0.329153 - 0.248509i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (-3.79 + 1.60i)T \)
good5 \( 1 + (2.87 - 2.87i)T - 5iT^{2} \)
7 \( 1 + (0.652 + 0.652i)T + 7iT^{2} \)
11 \( 1 + (1.60 + 1.60i)T + 11iT^{2} \)
13 \( 1 + 3.57T + 13T^{2} \)
19 \( 1 + 1.72iT - 19T^{2} \)
23 \( 1 + (-3.60 - 3.60i)T + 23iT^{2} \)
29 \( 1 + (1.55 - 1.55i)T - 29iT^{2} \)
31 \( 1 + (-2.92 + 2.92i)T - 31iT^{2} \)
37 \( 1 + (-4.21 + 4.21i)T - 37iT^{2} \)
41 \( 1 + (-3.57 - 3.57i)T + 41iT^{2} \)
43 \( 1 + 5.23iT - 43T^{2} \)
47 \( 1 + 6.96T + 47T^{2} \)
53 \( 1 + 2.69iT - 53T^{2} \)
59 \( 1 + 7.39iT - 59T^{2} \)
61 \( 1 + (2.75 + 2.75i)T + 61iT^{2} \)
67 \( 1 + 12.0T + 67T^{2} \)
71 \( 1 + (1.07 - 1.07i)T - 71iT^{2} \)
73 \( 1 + (-0.823 + 0.823i)T - 73iT^{2} \)
79 \( 1 + (8.40 + 8.40i)T + 79iT^{2} \)
83 \( 1 - 6.38iT - 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + (10.2 - 10.2i)T - 97iT^{2} \)
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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

−10.19908122526127243918002025049, −9.471586454196869597501286414864, −8.109918693298998384895094178920, −7.41094967342263745285956733960, −6.80340799671962340797086942639, −5.62652302378174852418396958702, −4.55939709822180851461717612582, −3.48199856112909601541863618059, −2.82057711066780867507976187735, −0.24453528253130609023158144419, 1.19110540717660399035610086035, 2.91077616825027051832905563423, 4.34035607344737556951965070688, 4.93332228189295757663443722207, 5.92187309941630296780568769088, 7.24320882737735894816162515309, 7.81489742706897361793444219199, 8.545403087273764036327871299632, 9.515464264388311864093121494162, 10.42883591751331254444137463878

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