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Properties

Label 2-294-147.104-c1-0-13
Degree $2$
Conductor $294$
Sign $-0.682 + 0.730i$
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 + (−1.54 + 0.782i)3-s + (0.222 + 0.974i)4-s + (0.415 + 0.200i)5-s + (1.69 + 0.351i)6-s + (−2.57 + 0.627i)7-s + (0.433 − 0.900i)8-s + (1.77 − 2.41i)9-s + (−0.200 − 0.415i)10-s + (−2.98 − 2.37i)11-s + (−1.10 − 1.33i)12-s + (2.23 + 1.78i)13-s + (2.40 + 1.11i)14-s + (−0.799 + 0.0158i)15-s + (−0.900 + 0.433i)16-s + (1.05 − 4.60i)17-s + ⋯
L(s)  = 1  + (−0.552 − 0.440i)2-s + (−0.892 + 0.451i)3-s + (0.111 + 0.487i)4-s + (0.185 + 0.0895i)5-s + (0.692 + 0.143i)6-s + (−0.971 + 0.237i)7-s + (0.153 − 0.318i)8-s + (0.591 − 0.806i)9-s + (−0.0633 − 0.131i)10-s + (−0.899 − 0.717i)11-s + (−0.319 − 0.384i)12-s + (0.621 + 0.495i)13-s + (0.641 + 0.297i)14-s + (−0.206 + 0.00410i)15-s + (−0.225 + 0.108i)16-s + (0.254 − 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.682 + 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.682 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.682 + 0.730i$
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: \((2,\ 294,\ (\ :1/2),\ -0.682 + 0.730i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.123070 - 0.283302i\)
\(L(\frac12)\) \(\approx\) \(0.123070 - 0.283302i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.54 - 0.782i)T \)
7 \( 1 + (2.57 - 0.627i)T \)
good5 \( 1 + (-0.415 - 0.200i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (2.98 + 2.37i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-2.23 - 1.78i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-1.05 + 4.60i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + 7.86iT - 19T^{2} \)
23 \( 1 + (6.27 - 1.43i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (8.41 + 1.92i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + 2.28iT - 31T^{2} \)
37 \( 1 + (1.70 - 7.46i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-3.55 - 1.71i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-9.76 + 4.70i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-3.02 + 3.79i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (0.115 - 0.0264i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (5.93 - 2.85i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (1.62 + 0.371i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + 3.96T + 67T^{2} \)
71 \( 1 + (0.892 - 0.203i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (5.38 - 4.29i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 0.975T + 79T^{2} \)
83 \( 1 + (-8.84 - 11.0i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (5.73 + 7.18i)T + (-19.8 + 86.7i)T^{2} \)
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

−11.35519318787127013134790483541, −10.51399698438858851437666564031, −9.621095627941564670309544293078, −9.004455464818589052236221152737, −7.52188318963336898976122283544, −6.39887140537572191948289996631, −5.50640534269283143057668063969, −4.02577734690942670025771955540, −2.68755913838682105260668991444, −0.29519043162447330125513411179, 1.76509252620336810325432992709, 3.93889658894371778809075385458, 5.82602378010696445124117497652, 5.90349460754088443348372271729, 7.41191209389769880938553680098, 7.945708492969981716787203446520, 9.464985062576216587848618482641, 10.35573838629166711918477878400, 10.80760555230771599638716277916, 12.41794515354819535194820185190

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