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Properties

Label 2-297-99.31-c1-0-7
Degree $2$
Conductor $297$
Sign $-0.0823 + 0.996i$
Analytic cond. $2.37155$
Root an. cond. $1.53998$
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  + (−0.373 + 0.0794i)2-s + (−1.69 + 0.754i)4-s + (−1.20 − 0.256i)5-s + (0.104 − 0.994i)7-s + (1.19 − 0.865i)8-s + 0.472·10-s + (3.01 − 1.38i)11-s + (−4.33 − 4.80i)13-s + (0.0399 + 0.379i)14-s + (2.10 − 2.33i)16-s + (−1.5 − 4.61i)17-s + (−0.809 + 0.587i)19-s + (2.24 − 0.476i)20-s + (−1.01 + 0.756i)22-s + (2.30 − 3.99i)23-s + ⋯
L(s)  = 1  + (−0.264 + 0.0561i)2-s + (−0.846 + 0.377i)4-s + (−0.540 − 0.114i)5-s + (0.0395 − 0.375i)7-s + (0.421 − 0.305i)8-s + 0.149·10-s + (0.908 − 0.417i)11-s + (−1.20 − 1.33i)13-s + (0.0106 + 0.101i)14-s + (0.526 − 0.584i)16-s + (−0.363 − 1.11i)17-s + (−0.185 + 0.134i)19-s + (0.501 − 0.106i)20-s + (−0.216 + 0.161i)22-s + (0.481 − 0.833i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0823 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0823 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $-0.0823 + 0.996i$
Analytic conductor: \(2.37155\)
Root analytic conductor: \(1.53998\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 297,\ (\ :1/2),\ -0.0823 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.408743 - 0.443916i\)
\(L(\frac12)\) \(\approx\) \(0.408743 - 0.443916i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + (-3.01 + 1.38i)T \)
good2 \( 1 + (0.373 - 0.0794i)T + (1.82 - 0.813i)T^{2} \)
5 \( 1 + (1.20 + 0.256i)T + (4.56 + 2.03i)T^{2} \)
7 \( 1 + (-0.104 + 0.994i)T + (-6.84 - 1.45i)T^{2} \)
13 \( 1 + (4.33 + 4.80i)T + (-1.35 + 12.9i)T^{2} \)
17 \( 1 + (1.5 + 4.61i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-2.30 + 3.99i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.507 - 4.82i)T + (-28.3 - 6.02i)T^{2} \)
31 \( 1 + (-0.413 - 0.459i)T + (-3.24 + 30.8i)T^{2} \)
37 \( 1 + (-4.11 - 2.99i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (0.264 + 2.51i)T + (-40.1 + 8.52i)T^{2} \)
43 \( 1 + (0.927 + 1.60i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (10.9 + 4.85i)T + (31.4 + 34.9i)T^{2} \)
53 \( 1 + (1.26 - 3.88i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (1.47 - 0.658i)T + (39.4 - 43.8i)T^{2} \)
61 \( 1 + (-7.26 + 8.06i)T + (-6.37 - 60.6i)T^{2} \)
67 \( 1 + (3 - 5.19i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.899 - 2.76i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.118 - 0.0857i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (10.3 - 2.19i)T + (72.1 - 32.1i)T^{2} \)
83 \( 1 + (-6.53 + 7.25i)T + (-8.67 - 82.5i)T^{2} \)
89 \( 1 - 6.76T + 89T^{2} \)
97 \( 1 + (5.86 - 1.24i)T + (88.6 - 39.4i)T^{2} \)
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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

−11.62586923718619513329619811787, −10.42096882991128533204853514922, −9.560832692113447282815666180326, −8.611396320797135280439396854211, −7.78376383726298678119775174758, −6.89170599144496764549293122662, −5.22664590040446573020865386283, −4.29802689798368470000732328330, −3.09780174098726394511677564333, −0.50861976413599570306928276639, 1.85564955673627588586794145733, 3.94208229332334481488601562344, 4.70165493425038130331211362870, 6.08853614377678013344415953103, 7.26113594927741257327473153849, 8.338432500407788032356748659039, 9.348357316957084959494008823390, 9.796539109539348182685909027416, 11.18348466812266526805719964446, 11.86934119184326178180027402212

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