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Properties

Label 2-804-268.267-c1-0-29
Degree $2$
Conductor $804$
Sign $-0.0235 - 0.999i$
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  + (0.271 + 1.38i)2-s + 3-s + (−1.85 + 0.754i)4-s − 0.299i·5-s + (0.271 + 1.38i)6-s + 1.88·7-s + (−1.55 − 2.36i)8-s + 9-s + (0.416 − 0.0814i)10-s + 2.47·11-s + (−1.85 + 0.754i)12-s + 1.99i·13-s + (0.511 + 2.60i)14-s − 0.299i·15-s + (2.86 − 2.79i)16-s + 0.452·17-s + ⋯
L(s)  = 1  + (0.192 + 0.981i)2-s + 0.577·3-s + (−0.926 + 0.377i)4-s − 0.134i·5-s + (0.110 + 0.566i)6-s + 0.710·7-s + (−0.548 − 0.836i)8-s + 0.333·9-s + (0.131 − 0.0257i)10-s + 0.744·11-s + (−0.534 + 0.217i)12-s + 0.554i·13-s + (0.136 + 0.697i)14-s − 0.0774i·15-s + (0.715 − 0.698i)16-s + 0.109·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0235 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0235 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.0235 - 0.999i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (535, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ -0.0235 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45669 + 1.49137i\)
\(L(\frac12)\) \(\approx\) \(1.45669 + 1.49137i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.271 - 1.38i)T \)
3 \( 1 - T \)
67 \( 1 + (2.90 - 7.65i)T \)
good5 \( 1 + 0.299iT - 5T^{2} \)
7 \( 1 - 1.88T + 7T^{2} \)
11 \( 1 - 2.47T + 11T^{2} \)
13 \( 1 - 1.99iT - 13T^{2} \)
17 \( 1 - 0.452T + 17T^{2} \)
19 \( 1 - 5.21iT - 19T^{2} \)
23 \( 1 - 0.807iT - 23T^{2} \)
29 \( 1 - 9.00T + 29T^{2} \)
31 \( 1 + 8.22T + 31T^{2} \)
37 \( 1 - 1.31T + 37T^{2} \)
41 \( 1 - 6.67iT - 41T^{2} \)
43 \( 1 + 4.44T + 43T^{2} \)
47 \( 1 + 2.51iT - 47T^{2} \)
53 \( 1 - 0.990iT - 53T^{2} \)
59 \( 1 + 7.56iT - 59T^{2} \)
61 \( 1 + 9.06iT - 61T^{2} \)
71 \( 1 + 0.0862iT - 71T^{2} \)
73 \( 1 - 1.91T + 73T^{2} \)
79 \( 1 + 8.30T + 79T^{2} \)
83 \( 1 + 0.827iT - 83T^{2} \)
89 \( 1 - 14.5T + 89T^{2} \)
97 \( 1 + 6.11iT - 97T^{2} \)
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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

−10.17611635523044158850484485223, −9.352198679078745397445448338765, −8.558294176528166512925063361229, −8.001313665584536329375111554535, −7.03163111623298162392637195088, −6.25126111278760367100692946359, −5.08419247557669900828117473125, −4.28023792157894009611131912807, −3.29159134021812515277685840432, −1.50925682611683350861064065841, 1.11514113643313398861056736268, 2.41197580599474770302676181716, 3.36655068944601587254822882515, 4.45431048405198661263026161603, 5.22629301399780743976567371741, 6.55931665797709763227224391495, 7.67796437476458184058706255739, 8.717215168833003384236030617062, 9.099209550757467471442733702516, 10.23023455926840830489382647467

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