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Properties

Label 2-297-9.7-c1-0-6
Degree $2$
Conductor $297$
Sign $0.199 + 0.979i$
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  + (1.07 − 1.86i)2-s + (−1.32 − 2.29i)4-s + (1.81 + 3.13i)5-s + (1.13 − 1.96i)7-s − 1.39·8-s + 7.81·10-s + (0.5 − 0.866i)11-s + (−0.619 − 1.07i)13-s + (−2.44 − 4.23i)14-s + (1.14 − 1.98i)16-s − 5.69·17-s − 2.89·19-s + (4.79 − 8.30i)20-s + (−1.07 − 1.86i)22-s + (2.95 + 5.12i)23-s + ⋯
L(s)  = 1  + (0.762 − 1.32i)2-s + (−0.661 − 1.14i)4-s + (0.810 + 1.40i)5-s + (0.429 − 0.743i)7-s − 0.492·8-s + 2.47·10-s + (0.150 − 0.261i)11-s + (−0.171 − 0.297i)13-s + (−0.654 − 1.13i)14-s + (0.286 − 0.495i)16-s − 1.38·17-s − 0.664·19-s + (1.07 − 1.85i)20-s + (−0.229 − 0.398i)22-s + (0.616 + 1.06i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.199 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.199 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.199 + 0.979i$
Analytic conductor: \(2.37155\)
Root analytic conductor: \(1.53998\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 297,\ (\ :1/2),\ 0.199 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64812 - 1.34614i\)
\(L(\frac12)\) \(\approx\) \(1.64812 - 1.34614i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-1.07 + 1.86i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.81 - 3.13i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.13 + 1.96i)T + (-3.5 - 6.06i)T^{2} \)
13 \( 1 + (0.619 + 1.07i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.69T + 17T^{2} \)
19 \( 1 + 2.89T + 19T^{2} \)
23 \( 1 + (-2.95 - 5.12i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.75 - 3.03i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.25 - 2.17i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.333T + 37T^{2} \)
41 \( 1 + (4.98 + 8.63i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.57 + 6.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.34 - 7.53i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6.16T + 53T^{2} \)
59 \( 1 + (-1.45 - 2.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.13 - 5.43i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.68 + 8.10i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.1T + 71T^{2} \)
73 \( 1 - 4.31T + 73T^{2} \)
79 \( 1 + (0.708 - 1.22i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.37 - 2.38i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 4.77T + 89T^{2} \)
97 \( 1 + (-1.27 + 2.21i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.19029948207697626300386681301, −10.88685481694356970875487734617, −10.23488320112372532813205161353, −9.149262142585097805871474416931, −7.46014804044529639177728775403, −6.53229609741090188743348525931, −5.23155115152209177498420108392, −3.96627423549327335339968812075, −2.91398737336228229182797804166, −1.81128628278821178795420221654, 2.01036875555960058589589424130, 4.47646889689203125040778213404, 4.90166196792652327301788904913, 5.97380898226423146735120724188, 6.73828456672548271480983972865, 8.244622329069013606756594818160, 8.732161081712882378394782459645, 9.723410195496206805607703490271, 11.24533798912850158042080381749, 12.47739469581961092359538198208

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