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Properties

Label 2-297-99.58-c1-0-0
Degree $2$
Conductor $297$
Sign $0.436 - 0.899i$
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  + (−1.41 − 0.628i)2-s + (0.261 + 0.290i)4-s + (−0.540 + 0.240i)5-s + (−0.706 + 0.150i)7-s + (0.768 + 2.36i)8-s + 0.915·10-s + (−3.10 + 1.16i)11-s + (0.164 + 1.56i)13-s + (1.09 + 0.232i)14-s + (0.483 − 4.60i)16-s + (3.71 + 2.69i)17-s + (0.775 + 2.38i)19-s + (−0.211 − 0.0940i)20-s + (5.11 + 0.300i)22-s + (−2.22 + 3.85i)23-s + ⋯
L(s)  = 1  + (−0.998 − 0.444i)2-s + (0.130 + 0.145i)4-s + (−0.241 + 0.107i)5-s + (−0.266 + 0.0567i)7-s + (0.271 + 0.836i)8-s + 0.289·10-s + (−0.935 + 0.352i)11-s + (0.0456 + 0.434i)13-s + (0.291 + 0.0620i)14-s + (0.120 − 1.15i)16-s + (0.901 + 0.654i)17-s + (0.177 + 0.547i)19-s + (−0.0472 − 0.0210i)20-s + (1.09 + 0.0641i)22-s + (−0.464 + 0.803i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.436 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.436 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.436 - 0.899i$
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: \((2,\ 297,\ (\ :1/2),\ 0.436 - 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.403012 + 0.252516i\)
\(L(\frac12)\) \(\approx\) \(0.403012 + 0.252516i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.10 - 1.16i)T \)
good2 \( 1 + (1.41 + 0.628i)T + (1.33 + 1.48i)T^{2} \)
5 \( 1 + (0.540 - 0.240i)T + (3.34 - 3.71i)T^{2} \)
7 \( 1 + (0.706 - 0.150i)T + (6.39 - 2.84i)T^{2} \)
13 \( 1 + (-0.164 - 1.56i)T + (-12.7 + 2.70i)T^{2} \)
17 \( 1 + (-3.71 - 2.69i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.775 - 2.38i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (2.22 - 3.85i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.82 + 1.45i)T + (26.4 - 11.7i)T^{2} \)
31 \( 1 + (-0.954 - 9.07i)T + (-30.3 + 6.44i)T^{2} \)
37 \( 1 + (-0.893 + 2.75i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.12 + 0.239i)T + (37.4 + 16.6i)T^{2} \)
43 \( 1 + (-2.10 - 3.64i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.152 + 0.169i)T + (-4.91 - 46.7i)T^{2} \)
53 \( 1 + (4.61 - 3.35i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (4.77 + 5.30i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (0.476 - 4.53i)T + (-59.6 - 12.6i)T^{2} \)
67 \( 1 + (-4.04 + 7.00i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (9.95 + 7.23i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.78 + 14.7i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (13.3 + 5.94i)T + (52.8 + 58.7i)T^{2} \)
83 \( 1 + (0.726 - 6.91i)T + (-81.1 - 17.2i)T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 + (-12.9 - 5.76i)T + (64.9 + 72.0i)T^{2} \)
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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

−11.76025197880904804674866572502, −10.69380848777246416744087254774, −10.07562641623576271057163019965, −9.272374425073652221048334610704, −8.149926998053267018289362164545, −7.53600187686861884175768329983, −6.00334882623448347580755581453, −4.84203869893726838497429109557, −3.21182932466695042573168020921, −1.62876219064935473984843638185, 0.50341543795173691847549810311, 2.92324195812257398510869556306, 4.40697847781914967461035667835, 5.81801396934769171154164141772, 7.01496968449918782129055232548, 7.977612835752765739937617236840, 8.487158414405044612571664126456, 9.784192958484748507850560825664, 10.22337779848379931473008991991, 11.46593839087442896034255775287

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