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Properties

Label 2-294-147.104-c1-0-1
Degree $2$
Conductor $294$
Sign $0.364 - 0.931i$
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 + (−1.59 − 0.676i)3-s + (0.222 + 0.974i)4-s + (−2.59 − 1.25i)5-s + (0.824 + 1.52i)6-s + (−2.01 − 1.71i)7-s + (0.433 − 0.900i)8-s + (2.08 + 2.15i)9-s + (1.25 + 2.59i)10-s + (3.63 + 2.90i)11-s + (0.304 − 1.70i)12-s + (−1.88 − 1.50i)13-s + (0.501 + 2.59i)14-s + (3.29 + 3.75i)15-s + (−0.900 + 0.433i)16-s + (−1.66 + 7.30i)17-s + ⋯
L(s)  = 1  + (−0.552 − 0.440i)2-s + (−0.920 − 0.390i)3-s + (0.111 + 0.487i)4-s + (−1.16 − 0.559i)5-s + (0.336 + 0.621i)6-s + (−0.760 − 0.649i)7-s + (0.153 − 0.318i)8-s + (0.694 + 0.719i)9-s + (0.395 + 0.821i)10-s + (1.09 + 0.874i)11-s + (0.0880 − 0.492i)12-s + (−0.522 − 0.416i)13-s + (0.133 + 0.694i)14-s + (0.851 + 0.969i)15-s + (−0.225 + 0.108i)16-s + (−0.404 + 1.77i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.364 - 0.931i$
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: \((2,\ 294,\ (\ :1/2),\ 0.364 - 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.198099 + 0.135227i\)
\(L(\frac12)\) \(\approx\) \(0.198099 + 0.135227i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.59 + 0.676i)T \)
7 \( 1 + (2.01 + 1.71i)T \)
good5 \( 1 + (2.59 + 1.25i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (-3.63 - 2.90i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (1.88 + 1.50i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (1.66 - 7.30i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 - 2.97iT - 19T^{2} \)
23 \( 1 + (-4.11 + 0.938i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (5.21 + 1.18i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 - 6.11iT - 31T^{2} \)
37 \( 1 + (0.628 - 2.75i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-0.953 - 0.459i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-4.51 + 2.17i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (0.950 - 1.19i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (10.0 - 2.29i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (-7.30 + 3.51i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (11.6 + 2.66i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + 12.2T + 67T^{2} \)
71 \( 1 + (12.9 - 2.95i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (0.233 - 0.186i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 8.33T + 79T^{2} \)
83 \( 1 + (-8.04 - 10.0i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (1.18 + 1.48i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 5.44iT - 97T^{2} \)
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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

−12.14194359497019998574534210612, −10.98575633829067924067813261248, −10.30830552435079513880460718618, −9.220359029300817262750076092448, −8.000523726122537460076972242509, −7.21505024478188886974427865489, −6.28542429154100152232565048185, −4.56394946612173122773733737197, −3.73144257814871479934567890549, −1.40013824821974513104311404243, 0.25789117648787665487834063561, 3.16385921200215533450863879529, 4.50473157315980889453224084280, 5.81242385716733294451766439975, 6.81262133369422403566059591263, 7.40481583415651284261592955040, 9.169885141315996325065241795442, 9.349091009347545204435737646501, 10.89428846834670496579717186613, 11.51318106378540771279014837112

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