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Properties

Label 2-294-147.104-c1-0-0
Degree $2$
Conductor $294$
Sign $-0.867 + 0.497i$
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 + (−1.32 + 1.11i)3-s + (0.222 + 0.974i)4-s + (−3.37 − 1.62i)5-s + (−1.73 + 0.0435i)6-s + (−0.0659 + 2.64i)7-s + (−0.433 + 0.900i)8-s + (0.519 − 2.95i)9-s + (−1.62 − 3.37i)10-s + (−1.00 − 0.803i)11-s + (−1.38 − 1.04i)12-s + (−5.22 − 4.16i)13-s + (−1.70 + 2.02i)14-s + (6.28 − 1.60i)15-s + (−0.900 + 0.433i)16-s + (−0.308 + 1.35i)17-s + ⋯
L(s)  = 1  + (0.552 + 0.440i)2-s + (−0.765 + 0.642i)3-s + (0.111 + 0.487i)4-s + (−1.50 − 0.726i)5-s + (−0.706 + 0.0177i)6-s + (−0.0249 + 0.999i)7-s + (−0.153 + 0.318i)8-s + (0.173 − 0.984i)9-s + (−0.514 − 1.06i)10-s + (−0.303 − 0.242i)11-s + (−0.398 − 0.301i)12-s + (−1.44 − 1.15i)13-s + (−0.454 + 0.541i)14-s + (1.62 − 0.413i)15-s + (−0.225 + 0.108i)16-s + (−0.0747 + 0.327i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.867 + 0.497i$
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: \((2,\ 294,\ (\ :1/2),\ -0.867 + 0.497i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0644852 - 0.242304i\)
\(L(\frac12)\) \(\approx\) \(0.0644852 - 0.242304i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.32 - 1.11i)T \)
7 \( 1 + (0.0659 - 2.64i)T \)
good5 \( 1 + (3.37 + 1.62i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (1.00 + 0.803i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (5.22 + 4.16i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (0.308 - 1.35i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 - 3.51iT - 19T^{2} \)
23 \( 1 + (6.90 - 1.57i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-1.86 - 0.426i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 - 2.07iT - 31T^{2} \)
37 \( 1 + (1.39 - 6.10i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-4.71 - 2.27i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-9.86 + 4.75i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (1.70 - 2.14i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (14.1 - 3.22i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (2.50 - 1.20i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (0.509 + 0.116i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + 9.25T + 67T^{2} \)
71 \( 1 + (-9.61 + 2.19i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-6.25 + 4.98i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 0.520T + 79T^{2} \)
83 \( 1 + (5.43 + 6.81i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-3.62 - 4.54i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + 0.909iT - 97T^{2} \)
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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

−12.32872490878190259990959644028, −11.76348947135627888965388988948, −10.63839534560078641029538430403, −9.466691920807546235783454453641, −8.242247106831826835028931545615, −7.65197538916245610027636143668, −6.07122299908982143087002355127, −5.19821189164580202073222282369, −4.39286975048049044249817795873, −3.20058636099949165475265285335, 0.16239757101005030999019189674, 2.44053255303087753124971815005, 4.09486743687396872401427958826, 4.74884904795076866715528489803, 6.47799723170091932137934116916, 7.26889107756393991627336397784, 7.78467607714389781350966208910, 9.720462764991687754284179491828, 10.79153416512118445363510584744, 11.32518105873153335502064974818

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