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Properties

Label 2-804-201.200-c1-0-10
Degree $2$
Conductor $804$
Sign $0.604 - 0.796i$
Analytic cond. $6.41997$
Root an. cond. $2.53376$
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  + (0.760 + 1.55i)3-s + 3.48·5-s − 1.71i·7-s + (−1.84 + 2.36i)9-s − 0.624·11-s + 2.24i·13-s + (2.65 + 5.42i)15-s + 3.10i·17-s + 5.86·19-s + (2.67 − 1.30i)21-s − 1.75i·23-s + 7.17·25-s + (−5.08 − 1.06i)27-s + 0.0875i·29-s + 0.222i·31-s + ⋯
L(s)  = 1  + (0.439 + 0.898i)3-s + 1.56·5-s − 0.649i·7-s + (−0.614 + 0.789i)9-s − 0.188·11-s + 0.623i·13-s + (0.685 + 1.40i)15-s + 0.753i·17-s + 1.34·19-s + (0.583 − 0.285i)21-s − 0.365i·23-s + 1.43·25-s + (−0.978 − 0.205i)27-s + 0.0162i·29-s + 0.0399i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.604 - 0.796i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ 0.604 - 0.796i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05248 + 1.01909i\)
\(L(\frac12)\) \(\approx\) \(2.05248 + 1.01909i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.760 - 1.55i)T \)
67 \( 1 + (3.68 + 7.30i)T \)
good5 \( 1 - 3.48T + 5T^{2} \)
7 \( 1 + 1.71iT - 7T^{2} \)
11 \( 1 + 0.624T + 11T^{2} \)
13 \( 1 - 2.24iT - 13T^{2} \)
17 \( 1 - 3.10iT - 17T^{2} \)
19 \( 1 - 5.86T + 19T^{2} \)
23 \( 1 + 1.75iT - 23T^{2} \)
29 \( 1 - 0.0875iT - 29T^{2} \)
31 \( 1 - 0.222iT - 31T^{2} \)
37 \( 1 - 2.86T + 37T^{2} \)
41 \( 1 + 5.27T + 41T^{2} \)
43 \( 1 + 7.30iT - 43T^{2} \)
47 \( 1 - 1.62iT - 47T^{2} \)
53 \( 1 + 7.57T + 53T^{2} \)
59 \( 1 + 4.77iT - 59T^{2} \)
61 \( 1 - 11.6iT - 61T^{2} \)
71 \( 1 - 5.72iT - 71T^{2} \)
73 \( 1 - 5.99T + 73T^{2} \)
79 \( 1 + 8.28iT - 79T^{2} \)
83 \( 1 + 8.79iT - 83T^{2} \)
89 \( 1 - 13.2iT - 89T^{2} \)
97 \( 1 + 18.0iT - 97T^{2} \)
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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

−10.28389064319134203286460920330, −9.578588070765998216455178906620, −9.010647662707028408642759672001, −7.968680549321664870558919438210, −6.84616503106090791658428931419, −5.81838362827492535429862638497, −5.06070534280718272220855479120, −3.99758929540464435479458136230, −2.83790019265556508679967103561, −1.66864868411439089838275960143, 1.26670611295049532769555624826, 2.40901939926259532424781470331, 3.14811728040670759031550609088, 5.17164993008294850300882155250, 5.74925164858299322447754451392, 6.59928898212786784948832677618, 7.55892714123275292831452856102, 8.451882632278239600565423652481, 9.503022149990643971924605212981, 9.684566479211237684788897105781

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