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Properties

Label 2-297-99.97-c1-0-7
Degree $2$
Conductor $297$
Sign $0.577 + 0.816i$
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  + (0.255 + 0.283i)2-s + (0.193 − 1.84i)4-s + (0.827 − 0.918i)5-s + (−0.913 + 0.406i)7-s + (1.19 − 0.865i)8-s + 0.472·10-s + (−2.70 − 1.91i)11-s + (6.33 − 1.34i)13-s + (−0.348 − 0.155i)14-s + (−3.07 − 0.654i)16-s + (−1.5 − 4.61i)17-s + (−0.809 + 0.587i)19-s + (−1.53 − 1.70i)20-s + (−0.146 − 1.25i)22-s + (2.30 + 3.99i)23-s + ⋯
L(s)  = 1  + (0.180 + 0.200i)2-s + (0.0969 − 0.921i)4-s + (0.369 − 0.410i)5-s + (−0.345 + 0.153i)7-s + (0.421 − 0.305i)8-s + 0.149·10-s + (−0.815 − 0.578i)11-s + (1.75 − 0.373i)13-s + (−0.0932 − 0.0415i)14-s + (−0.769 − 0.163i)16-s + (−0.363 − 1.11i)17-s + (−0.185 + 0.134i)19-s + (−0.342 − 0.380i)20-s + (−0.0312 − 0.268i)22-s + (0.481 + 0.833i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(2.37155\)
Root analytic conductor: \(1.53998\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 297,\ (\ :1/2),\ 0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31039 - 0.678145i\)
\(L(\frac12)\) \(\approx\) \(1.31039 - 0.678145i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.70 + 1.91i)T \)
good2 \( 1 + (-0.255 - 0.283i)T + (-0.209 + 1.98i)T^{2} \)
5 \( 1 + (-0.827 + 0.918i)T + (-0.522 - 4.97i)T^{2} \)
7 \( 1 + (0.913 - 0.406i)T + (4.68 - 5.20i)T^{2} \)
13 \( 1 + (-6.33 + 1.34i)T + (11.8 - 5.28i)T^{2} \)
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 \( 1 + (-4.43 + 1.97i)T + (19.4 - 21.5i)T^{2} \)
31 \( 1 + (0.604 - 0.128i)T + (28.3 - 12.6i)T^{2} \)
37 \( 1 + (-4.11 - 2.99i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-2.30 - 1.02i)T + (27.4 + 30.4i)T^{2} \)
43 \( 1 + (0.927 - 1.60i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.24 - 11.8i)T + (-45.9 + 9.77i)T^{2} \)
53 \( 1 + (1.26 - 3.88i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-0.169 + 1.60i)T + (-57.7 - 12.2i)T^{2} \)
61 \( 1 + (10.6 + 2.25i)T + (55.7 + 24.8i)T^{2} \)
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 \( 1 + (-7.06 - 7.84i)T + (-8.25 + 78.5i)T^{2} \)
83 \( 1 + (9.55 + 2.03i)T + (75.8 + 33.7i)T^{2} \)
89 \( 1 - 6.76T + 89T^{2} \)
97 \( 1 + (-4.01 - 4.45i)T + (-10.1 + 96.4i)T^{2} \)
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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

−11.30255594766115160054473229556, −10.81924220679157519623877246626, −9.662455527653611132415820971939, −8.949934021101018249860712627022, −7.74499388224996139605978064741, −6.34821450145739813590314455584, −5.73191512180616323252228771698, −4.71478020238401339815520848352, −3.02377680108755967950666369772, −1.16524693250119384680319082567, 2.19214868205228066414876469208, 3.45892364458785405794320964675, 4.51298057008392313957410106497, 6.15288202498077948082343695202, 6.93909597396421758829258013687, 8.195729192549163644329308282772, 8.860969953078241159178958575481, 10.38481541691586738205469884848, 10.83744575264338527736676402989, 11.97325228356527272426219917433

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