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Properties

Label 2-294-147.104-c1-0-9
Degree $2$
Conductor $294$
Sign $0.983 - 0.183i$
Analytic cond. $2.34760$
Root an. cond. $1.53218$
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  + (−0.781 − 0.623i)2-s + (1.38 + 1.03i)3-s + (0.222 + 0.974i)4-s + (1.24 + 0.599i)5-s + (−0.438 − 1.67i)6-s + (2.35 − 1.21i)7-s + (0.433 − 0.900i)8-s + (0.850 + 2.87i)9-s + (−0.599 − 1.24i)10-s + (−4.74 − 3.78i)11-s + (−0.701 + 1.58i)12-s + (3.67 + 2.93i)13-s + (−2.59 − 0.515i)14-s + (1.10 + 2.12i)15-s + (−0.900 + 0.433i)16-s + (−1.02 + 4.50i)17-s + ⋯
L(s)  = 1  + (−0.552 − 0.440i)2-s + (0.801 + 0.598i)3-s + (0.111 + 0.487i)4-s + (0.557 + 0.268i)5-s + (−0.179 − 0.684i)6-s + (0.888 − 0.459i)7-s + (0.153 − 0.318i)8-s + (0.283 + 0.958i)9-s + (−0.189 − 0.393i)10-s + (−1.43 − 1.14i)11-s + (−0.202 + 0.457i)12-s + (1.01 + 0.813i)13-s + (−0.693 − 0.137i)14-s + (0.285 + 0.548i)15-s + (−0.225 + 0.108i)16-s + (−0.249 + 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.983 - 0.183i$
Analytic conductor: \(2.34760\)
Root analytic conductor: \(1.53218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1/2),\ 0.983 - 0.183i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44294 + 0.133585i\)
\(L(\frac12)\) \(\approx\) \(1.44294 + 0.133585i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.781 + 0.623i)T \)
3 \( 1 + (-1.38 - 1.03i)T \)
7 \( 1 + (-2.35 + 1.21i)T \)
good5 \( 1 + (-1.24 - 0.599i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (4.74 + 3.78i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-3.67 - 2.93i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (1.02 - 4.50i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + 0.830iT - 19T^{2} \)
23 \( 1 + (-6.70 + 1.53i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (5.02 + 1.14i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 - 3.29iT - 31T^{2} \)
37 \( 1 + (-0.766 + 3.35i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (3.21 + 1.54i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-2.30 + 1.10i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-1.68 + 2.10i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (8.03 - 1.83i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (-8.71 + 4.19i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (6.90 + 1.57i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + 9.57T + 67T^{2} \)
71 \( 1 + (1.12 - 0.257i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (10.6 - 8.47i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 15.0T + 79T^{2} \)
83 \( 1 + (2.44 + 3.07i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-5.74 - 7.20i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + 14.5iT - 97T^{2} \)
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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

−11.12722400149537925628706225273, −10.89237903154627754867818490100, −10.07296627920535825870550270494, −8.766386077707139031605442687397, −8.407766673064407331641463817602, −7.29712235332028773075865957422, −5.73256220509886557849261378954, −4.35316026371737145162744991376, −3.13581242538608271069399318176, −1.83649477905241219685721931508, 1.54800099617992628889111651223, 2.77609074070589370763404872916, 4.89932640988086369148388359632, 5.81853507287406580488135692244, 7.31298703727789839197752525255, 7.82689998359003620818878484194, 8.831411680703758872038070206557, 9.546463863324310696839426779538, 10.63377587360675580706419346918, 11.72179590183506551061752392232

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