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Properties

Label 2-294-21.5-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: $\chi_{294} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1/2),\ 0.750 + 0.660i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88752 - 0.712648i\)
\(L(\frac12)\) \(\approx\) \(1.88752 - 0.712648i\)
\(L(\frac{3}{2})\) 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

−11.99342062561895787039704912156, −11.10917058524652701183405692845, −9.325126340474154473454784137562, −8.854502309571242747888304441404, −7.57467285040792752157493828159, −6.78986953738778758778480706442, −5.65471065119823300525937869832, −4.74240858001966720817261091805, −3.04778380868910875225824140277, −1.52033571182719440134516656783, 2.43931212600039648022322757443, 3.38394233096698559790956340543, 4.62000898740961187900118982857, 5.74453715200082488414562472972, 6.70430516509750544869725474696, 8.175659511678691741603651479724, 9.353490914889996467775521331092, 10.31982833411183174147518942130, 10.74797562061856866060339409905, 11.66069882800316157623316302072

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