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Properties

Label 2-804-67.64-c1-0-11
Degree $2$
Conductor $804$
Sign $-0.984 - 0.172i$
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  + (0.142 − 0.989i)3-s + (−0.536 − 1.17i)5-s + (−0.878 − 0.257i)7-s + (−0.959 − 0.281i)9-s + (−1.94 − 4.25i)11-s + (−2.64 + 3.05i)13-s + (−1.23 + 0.363i)15-s + (−5.31 + 3.41i)17-s + (1.25 − 0.367i)19-s + (−0.380 + 0.832i)21-s + (−0.397 + 2.76i)23-s + (2.18 − 2.51i)25-s + (−0.415 + 0.909i)27-s − 4.58·29-s + (1.11 + 1.28i)31-s + ⋯
L(s)  = 1  + (0.0821 − 0.571i)3-s + (−0.239 − 0.525i)5-s + (−0.331 − 0.0974i)7-s + (−0.319 − 0.0939i)9-s + (−0.586 − 1.28i)11-s + (−0.733 + 0.846i)13-s + (−0.319 + 0.0939i)15-s + (−1.28 + 0.828i)17-s + (0.287 − 0.0843i)19-s + (−0.0829 + 0.181i)21-s + (−0.0828 + 0.576i)23-s + (0.436 − 0.503i)25-s + (−0.0799 + 0.175i)27-s − 0.851·29-s + (0.199 + 0.230i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.984 - 0.172i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ -0.984 - 0.172i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0365819 + 0.420601i\)
\(L(\frac12)\) \(\approx\) \(0.0365819 + 0.420601i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.142 + 0.989i)T \)
67 \( 1 + (-5.64 - 5.92i)T \)
good5 \( 1 + (0.536 + 1.17i)T + (-3.27 + 3.77i)T^{2} \)
7 \( 1 + (0.878 + 0.257i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (1.94 + 4.25i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (2.64 - 3.05i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (5.31 - 3.41i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (-1.25 + 0.367i)T + (15.9 - 10.2i)T^{2} \)
23 \( 1 + (0.397 - 2.76i)T + (-22.0 - 6.47i)T^{2} \)
29 \( 1 + 4.58T + 29T^{2} \)
31 \( 1 + (-1.11 - 1.28i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + 3.19T + 37T^{2} \)
41 \( 1 + (-9.53 + 6.12i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (9.19 - 5.90i)T + (17.8 - 39.1i)T^{2} \)
47 \( 1 + (-1.17 + 8.18i)T + (-45.0 - 13.2i)T^{2} \)
53 \( 1 + (10.8 + 6.97i)T + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (-0.690 - 0.796i)T + (-8.39 + 58.3i)T^{2} \)
61 \( 1 + (5.72 - 12.5i)T + (-39.9 - 46.1i)T^{2} \)
71 \( 1 + (-0.713 - 0.458i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (-2.89 + 6.34i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (-4.15 + 4.79i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (2.02 + 4.42i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (0.858 + 5.97i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 + 15.7T + 97T^{2} \)
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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

−9.690840741141572542781510875315, −8.774012259002169095263514324405, −8.244793864146152930050289429850, −7.19629440021817801294031824907, −6.39245542693674502223718007781, −5.42291293839443739037687681542, −4.33086440165791915991619355307, −3.16462947186613904756107873006, −1.87963970318363866152526551588, −0.19212311643034213848367984176, 2.34523829673496145160192205027, 3.18358951031465814683683776102, 4.52709824862140708105015032001, 5.14717132184944566284216049669, 6.46192188492730310381323933302, 7.32591043704681117578834472037, 8.031723789160041469027293415699, 9.376293967959557476900631568327, 9.709449087504295961992601675514, 10.74002728514989991819654980720

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