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Properties

Label 2-297-99.32-c1-0-6
Degree $2$
Conductor $297$
Sign $0.800 + 0.598i$
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 − 1.73i)4-s + (0.813 + 0.469i)5-s + (2.87 − 1.65i)11-s + (−1.99 − 3.46i)16-s + (1.62 − 0.939i)20-s + (2.87 + 1.65i)23-s + (−2.05 − 3.56i)25-s + (−3.05 + 5.29i)31-s + 12.1·37-s − 6.63i·44-s + (−11.8 + 6.85i)47-s + (−3.5 + 6.06i)49-s + 11.8i·53-s + 3.11·55-s + (−12.6 − 7.32i)59-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)4-s + (0.363 + 0.210i)5-s + (0.866 − 0.500i)11-s + (−0.499 − 0.866i)16-s + (0.363 − 0.210i)20-s + (0.598 + 0.345i)23-s + (−0.411 − 0.713i)25-s + (−0.549 + 0.951i)31-s + 1.99·37-s − 1.00i·44-s + (−1.73 + 0.999i)47-s + (−0.5 + 0.866i)49-s + 1.62i·53-s + 0.420·55-s + (−1.65 − 0.953i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 + 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.800 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.800 + 0.598i$
Analytic conductor: \(2.37155\)
Root analytic conductor: \(1.53998\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (98, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 297,\ (\ :1/2),\ 0.800 + 0.598i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46407 - 0.486729i\)
\(L(\frac12)\) \(\approx\) \(1.46407 - 0.486729i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.87 + 1.65i)T \)
good2 \( 1 + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.813 - 0.469i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (3.5 - 6.06i)T^{2} \)
13 \( 1 + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (-2.87 - 1.65i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.05 - 5.29i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 12.1T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (11.8 - 6.85i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 11.8iT - 53T^{2} \)
59 \( 1 + (12.6 + 7.32i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.55 - 13.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.8iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 16.5iT - 89T^{2} \)
97 \( 1 + (0.0584 + 0.101i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.41385648539413257840844003934, −10.84103534238292678482085947163, −9.775529434110149219900547061564, −9.090280762694807270931781127515, −7.69372026952203388279995885158, −6.49932726519668840934007904059, −5.92553546542967692245899095663, −4.60853041874035947972088376335, −2.95127816312683059618374824394, −1.40501905323896177182896856443, 1.92744279389285753421860503538, 3.39780115576335479465276021166, 4.58869607639560251570757244828, 6.07981394777200980670435904824, 7.02184544327875127960369739447, 7.963237173800474893054123151582, 9.031022694687657522276005099475, 9.859529482838263651671773513681, 11.21107776861199286983061118382, 11.75540801316782644753043591458

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