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Properties

Label 2-294-147.104-c1-0-8
Degree $2$
Conductor $294$
Sign $-0.0318 - 0.999i$
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 + (−0.913 + 1.47i)3-s + (0.222 + 0.974i)4-s + (3.14 + 1.51i)5-s + (−1.63 + 0.580i)6-s + (2.47 − 0.926i)7-s + (−0.433 + 0.900i)8-s + (−1.32 − 2.68i)9-s + (1.51 + 3.14i)10-s + (−1.47 − 1.17i)11-s + (−1.63 − 0.563i)12-s + (−0.731 − 0.583i)13-s + (2.51 + 0.820i)14-s + (−5.11 + 3.24i)15-s + (−0.900 + 0.433i)16-s + (−1.24 + 5.43i)17-s + ⋯
L(s)  = 1  + (0.552 + 0.440i)2-s + (−0.527 + 0.849i)3-s + (0.111 + 0.487i)4-s + (1.40 + 0.678i)5-s + (−0.666 + 0.237i)6-s + (0.936 − 0.350i)7-s + (−0.153 + 0.318i)8-s + (−0.443 − 0.896i)9-s + (0.479 + 0.996i)10-s + (−0.445 − 0.355i)11-s + (−0.472 − 0.162i)12-s + (−0.202 − 0.161i)13-s + (0.672 + 0.219i)14-s + (−1.31 + 0.838i)15-s + (−0.225 + 0.108i)16-s + (−0.301 + 1.31i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0318 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0318 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.0318 - 0.999i$
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: \((2,\ 294,\ (\ :1/2),\ -0.0318 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27270 + 1.31384i\)
\(L(\frac12)\) \(\approx\) \(1.27270 + 1.31384i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.913 - 1.47i)T \)
7 \( 1 + (-2.47 + 0.926i)T \)
good5 \( 1 + (-3.14 - 1.51i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (1.47 + 1.17i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (0.731 + 0.583i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (1.24 - 5.43i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + 6.71iT - 19T^{2} \)
23 \( 1 + (5.84 - 1.33i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (0.519 + 0.118i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + 5.34iT - 31T^{2} \)
37 \( 1 + (-0.612 + 2.68i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-1.61 - 0.777i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (9.41 - 4.53i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-7.08 + 8.88i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-5.46 + 1.24i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (0.981 - 0.472i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-4.27 - 0.975i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + 9.15T + 67T^{2} \)
71 \( 1 + (5.70 - 1.30i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-1.01 + 0.808i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 15.3T + 79T^{2} \)
83 \( 1 + (-9.92 - 12.4i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-6.59 - 8.26i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 7.02iT - 97T^{2} \)
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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

−11.86650205013084187316219840071, −10.89424420248146018436038245640, −10.38632124593282587103444068691, −9.332375574968823212066419242852, −8.153846019717673915455774592579, −6.76214273151912238420714598121, −5.87604604360691437694048760317, −5.13750795556326287803379162220, −3.94598048828656882025750686976, −2.35269461611216637092757024505, 1.53978052265886482740449096277, 2.35876637246366574784402771979, 4.74959423298317842718543778496, 5.43056216519347693530517022243, 6.22858945481418237297105334898, 7.57619477000210259153608176364, 8.716724811616511456204482625561, 9.886510312867626770235663798026, 10.69441016151474899292839749154, 11.99974977607027956331131139028

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