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Properties

Label 2-297-99.2-c1-0-3
Degree $2$
Conductor $297$
Sign $-0.460 - 0.887i$
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  + (1.65 + 1.83i)2-s + (−0.429 + 4.09i)4-s + (1.51 + 1.36i)5-s + (−0.441 − 0.990i)7-s + (−4.22 + 3.07i)8-s + 5.05i·10-s + (−1.28 − 3.05i)11-s + (−0.231 − 1.08i)13-s + (1.09 − 2.44i)14-s + (−4.58 − 0.975i)16-s + (2.03 + 6.25i)17-s + (−4.19 − 5.76i)19-s + (−6.25 + 5.62i)20-s + (3.48 − 7.42i)22-s + (4.70 − 2.71i)23-s + ⋯
L(s)  = 1  + (1.16 + 1.29i)2-s + (−0.214 + 2.04i)4-s + (0.679 + 0.611i)5-s + (−0.166 − 0.374i)7-s + (−1.49 + 1.08i)8-s + 1.59i·10-s + (−0.388 − 0.921i)11-s + (−0.0641 − 0.301i)13-s + (0.291 − 0.654i)14-s + (−1.14 − 0.243i)16-s + (0.493 + 1.51i)17-s + (−0.961 − 1.32i)19-s + (−1.39 + 1.25i)20-s + (0.742 − 1.58i)22-s + (0.982 − 0.566i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 - 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $-0.460 - 0.887i$
Analytic conductor: \(2.37155\)
Root analytic conductor: \(1.53998\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 297,\ (\ :1/2),\ -0.460 - 0.887i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24226 + 2.04289i\)
\(L(\frac12)\) \(\approx\) \(1.24226 + 2.04289i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.28 + 3.05i)T \)
good2 \( 1 + (-1.65 - 1.83i)T + (-0.209 + 1.98i)T^{2} \)
5 \( 1 + (-1.51 - 1.36i)T + (0.522 + 4.97i)T^{2} \)
7 \( 1 + (0.441 + 0.990i)T + (-4.68 + 5.20i)T^{2} \)
13 \( 1 + (0.231 + 1.08i)T + (-11.8 + 5.28i)T^{2} \)
17 \( 1 + (-2.03 - 6.25i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.19 + 5.76i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (-4.70 + 2.71i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.76 - 3.01i)T + (19.4 - 21.5i)T^{2} \)
31 \( 1 + (-1.16 + 0.246i)T + (28.3 - 12.6i)T^{2} \)
37 \( 1 + (-0.780 - 0.567i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.00 - 0.449i)T + (27.4 + 30.4i)T^{2} \)
43 \( 1 + (-4.42 - 2.55i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.27 + 0.448i)T + (45.9 - 9.77i)T^{2} \)
53 \( 1 + (10.3 + 3.36i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-2.19 - 0.230i)T + (57.7 + 12.2i)T^{2} \)
61 \( 1 + (-1.20 + 5.67i)T + (-55.7 - 24.8i)T^{2} \)
67 \( 1 + (-0.496 - 0.860i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.12 - 1.98i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.486 - 0.669i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (4.53 - 4.07i)T + (8.25 - 78.5i)T^{2} \)
83 \( 1 + (2.14 + 0.456i)T + (75.8 + 33.7i)T^{2} \)
89 \( 1 - 6.19iT - 89T^{2} \)
97 \( 1 + (6.37 + 7.07i)T + (-10.1 + 96.4i)T^{2} \)
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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

−12.76195857899627506838371951719, −11.10447204688715162951311277106, −10.38672350338591638011654277165, −8.858549164809718482609628226814, −7.915819656180825507944685737815, −6.81601651684985470358061618519, −6.16906710740964354405535233038, −5.28526876158297372173675231018, −4.00175368894909497677299887249, −2.83623761276538754091722145658, 1.63183071588953032902239732226, 2.73399761475408125652209044597, 4.16663433707228270750189942751, 5.18302186781352145923534771807, 5.86682938611854059008618952432, 7.46127859736125443745733004556, 9.214003578928194880153776854673, 9.724231419284422714097455861925, 10.71079486114007057370992547053, 11.72741460696161274165400369626

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