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Properties

Label 2-294-21.17-c1-0-13
Degree $2$
Conductor $294$
Sign $-0.750 + 0.660i$
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  + (0.866 − 0.5i)2-s + (−0.448 − 1.67i)3-s + (0.499 − 0.866i)4-s + (−1.22 − 2.12i)5-s + (−1.22 − 1.22i)6-s − 0.999i·8-s + (−2.59 + 1.50i)9-s + (−2.12 − 1.22i)10-s + (−1.67 − 0.448i)12-s + 2.44i·13-s + (−3 + 3i)15-s + (−0.5 − 0.866i)16-s + (2.44 − 4.24i)17-s + (−1.5 + 2.59i)18-s + (−2.12 + 1.22i)19-s − 2.44·20-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.258 − 0.965i)3-s + (0.249 − 0.433i)4-s + (−0.547 − 0.948i)5-s + (−0.499 − 0.499i)6-s − 0.353i·8-s + (−0.866 + 0.5i)9-s + (−0.670 − 0.387i)10-s + (−0.482 − 0.129i)12-s + 0.679i·13-s + (−0.774 + 0.774i)15-s + (−0.125 − 0.216i)16-s + (0.594 − 1.02i)17-s + (−0.353 + 0.612i)18-s + (−0.486 + 0.280i)19-s − 0.547·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.750 + 0.660i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.750 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.750 + 0.660i$
Analytic conductor: \(2.34760\)
Root analytic conductor: \(1.53218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1/2),\ -0.750 + 0.660i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.495996 - 1.31369i\)
\(L(\frac12)\) \(\approx\) \(0.495996 - 1.31369i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (0.448 + 1.67i)T \)
7 \( 1 \)
good5 \( 1 + (1.22 + 2.12i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 + (-2.44 + 4.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.12 - 1.22i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.19 + 3i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.89T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-2.44 - 4.24i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.19 + 3i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.12 - 10.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.6 + 6.12i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-8.48 - 4.89i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.44T + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.89iT - 97T^{2} \)
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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

−11.71682059360920641616014116533, −10.92385886295460321596021673656, −9.475940657208469072640361536192, −8.440998109153861374612288280342, −7.44851319740481049702356618673, −6.39811595710393028385810578048, −5.24482610859698723303685189249, −4.27784272307937383260370266922, −2.58935657747147014352699697701, −0.942380357865129424766119560487, 3.03251427493489572676682153414, 3.79844069212401479768974505052, 5.06378716040843579740581560617, 6.05975789821906020938314333719, 7.14260682248202619571009941194, 8.200332604130035518570381957674, 9.366950827532195996454725750059, 10.80226637905720067043100146746, 10.83820987283200142375391327173, 12.10239247422441425531383382540

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