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Properties

Label 2-297-99.58-c1-0-4
Degree $2$
Conductor $297$
Sign $0.776 + 0.629i$
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  + (0.606 + 0.270i)2-s + (−1.04 − 1.15i)4-s + (−1.02 + 0.454i)5-s + (3.81 − 0.811i)7-s + (−0.730 − 2.24i)8-s − 0.741·10-s + (2.09 − 2.57i)11-s + (−0.279 − 2.66i)13-s + (2.53 + 0.539i)14-s + (−0.161 + 1.53i)16-s + (4.48 + 3.26i)17-s + (−1.58 − 4.88i)19-s + (1.58 + 0.707i)20-s + (1.96 − 0.993i)22-s + (1.05 − 1.83i)23-s + ⋯
L(s)  = 1  + (0.429 + 0.191i)2-s + (−0.521 − 0.579i)4-s + (−0.456 + 0.203i)5-s + (1.44 − 0.306i)7-s + (−0.258 − 0.794i)8-s − 0.234·10-s + (0.631 − 0.775i)11-s + (−0.0776 − 0.738i)13-s + (0.678 + 0.144i)14-s + (−0.0404 + 0.384i)16-s + (1.08 + 0.790i)17-s + (−0.363 − 1.11i)19-s + (0.355 + 0.158i)20-s + (0.419 − 0.211i)22-s + (0.220 − 0.381i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.776 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.776 + 0.629i$
Analytic conductor: \(2.37155\)
Root analytic conductor: \(1.53998\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 297,\ (\ :1/2),\ 0.776 + 0.629i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41056 - 0.499720i\)
\(L(\frac12)\) \(\approx\) \(1.41056 - 0.499720i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.09 + 2.57i)T \)
good2 \( 1 + (-0.606 - 0.270i)T + (1.33 + 1.48i)T^{2} \)
5 \( 1 + (1.02 - 0.454i)T + (3.34 - 3.71i)T^{2} \)
7 \( 1 + (-3.81 + 0.811i)T + (6.39 - 2.84i)T^{2} \)
13 \( 1 + (0.279 + 2.66i)T + (-12.7 + 2.70i)T^{2} \)
17 \( 1 + (-4.48 - 3.26i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.58 + 4.88i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-1.05 + 1.83i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.05 + 0.225i)T + (26.4 - 11.7i)T^{2} \)
31 \( 1 + (-0.0449 - 0.427i)T + (-30.3 + 6.44i)T^{2} \)
37 \( 1 + (2.69 - 8.27i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.85 + 0.606i)T + (37.4 + 16.6i)T^{2} \)
43 \( 1 + (-1.11 - 1.93i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (8.58 - 9.53i)T + (-4.91 - 46.7i)T^{2} \)
53 \( 1 + (-4.56 + 3.31i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.58 - 1.76i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (-0.426 + 4.06i)T + (-59.6 - 12.6i)T^{2} \)
67 \( 1 + (-4.87 + 8.44i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.11 - 3.71i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.60 - 11.1i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.12 - 1.83i)T + (52.8 + 58.7i)T^{2} \)
83 \( 1 + (-0.0503 + 0.479i)T + (-81.1 - 17.2i)T^{2} \)
89 \( 1 + 16.0T + 89T^{2} \)
97 \( 1 + (-7.32 - 3.26i)T + (64.9 + 72.0i)T^{2} \)
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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

−11.51553995098329666970717784594, −10.87078786119626656875706018838, −9.889111961879617540282954586883, −8.608672790013254818721827061847, −7.931288902894467985353401112378, −6.62057969218354342067325277970, −5.46372242117060549395773932583, −4.59691471009374425923941353847, −3.46995003144233962935523245983, −1.17879264106281141662220637510, 1.93765186519959295129898293107, 3.74597742432269817264807823269, 4.59177243255714172362937599569, 5.49026713406289715360964337831, 7.26025118135367346173496945016, 8.073464493097068339387689909560, 8.849298773220975935785640496618, 9.936075117411733674753084686920, 11.40051616143268824643828883225, 11.96068017280233130714465671978

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