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Properties

Label 2-297-11.3-c1-0-2
Degree $2$
Conductor $297$
Sign $-0.350 - 0.936i$
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  + (−1.85 + 1.35i)2-s + (1.01 − 3.12i)4-s + (−1.37 − 0.997i)5-s + (0.0641 − 0.197i)7-s + (0.910 + 2.80i)8-s + 3.90·10-s + (1.23 + 3.07i)11-s + (−1.11 + 0.812i)13-s + (0.147 + 0.453i)14-s + (−0.168 − 0.122i)16-s + (2.97 + 2.15i)17-s + (2.50 + 7.70i)19-s + (−4.50 + 3.27i)20-s + (−6.45 − 4.05i)22-s + 3.99·23-s + ⋯
L(s)  = 1  + (−1.31 + 0.955i)2-s + (0.507 − 1.56i)4-s + (−0.613 − 0.446i)5-s + (0.0242 − 0.0746i)7-s + (0.321 + 0.990i)8-s + 1.23·10-s + (0.372 + 0.928i)11-s + (−0.310 + 0.225i)13-s + (0.0394 + 0.121i)14-s + (−0.0420 − 0.0305i)16-s + (0.720 + 0.523i)17-s + (0.574 + 1.76i)19-s + (−1.00 + 0.732i)20-s + (−1.37 − 0.864i)22-s + 0.833·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $-0.350 - 0.936i$
Analytic conductor: \(2.37155\)
Root analytic conductor: \(1.53998\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 297,\ (\ :1/2),\ -0.350 - 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.308259 + 0.444287i\)
\(L(\frac12)\) \(\approx\) \(0.308259 + 0.444287i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.23 - 3.07i)T \)
good2 \( 1 + (1.85 - 1.35i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (1.37 + 0.997i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.0641 + 0.197i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.11 - 0.812i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.97 - 2.15i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.50 - 7.70i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 3.99T + 23T^{2} \)
29 \( 1 + (-0.167 + 0.515i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (7.95 - 5.77i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.381 - 1.17i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.88 - 8.87i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.92T + 43T^{2} \)
47 \( 1 + (2.42 + 7.47i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (5.30 - 3.85i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.43 + 7.48i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.12 - 0.816i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 5.85T + 67T^{2} \)
71 \( 1 + (-10.3 - 7.49i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.36 + 7.27i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.69 + 3.41i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-11.2 - 8.15i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 6.19T + 89T^{2} \)
97 \( 1 + (-5.68 + 4.12i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.09457702799359861465262617398, −10.73408377925903042497694007363, −9.845952111858013155297828476541, −9.131090034061559029876981211126, −8.050288163331780787633897146456, −7.54475278409060953536189965734, −6.50699756785550503046099631296, −5.30916682833453581796012429865, −3.85248637580233174241603465827, −1.40426966682744772065118061816, 0.69118331961868145371898814878, 2.63066072438966736696053013151, 3.58718046416989036539777291873, 5.41976131008459873515755921239, 7.14146331199181365533015306264, 7.74865770330220366333070890460, 9.017726490784166168194812651318, 9.405146380879681109608094647907, 10.78817380547146003719221317638, 11.18267538177204154078435594085

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