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Properties

Label 2-294-21.11-c2-0-7
Degree $2$
Conductor $294$
Sign $0.110 - 0.993i$
Analytic cond. $8.01091$
Root an. cond. $2.83035$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 + 0.707i)2-s + (−0.548 + 2.94i)3-s + (0.999 − 1.73i)4-s + (3.46 − 2i)5-s + (−1.41 − 3.99i)6-s + 2.82i·8-s + (−8.39 − 3.23i)9-s + (−2.82 + 4.89i)10-s + (8.57 + 4.94i)11-s + (4.56 + 3.89i)12-s + 12.7·13-s + (3.99 + 11.3i)15-s + (−2.00 − 3.46i)16-s + (27.7 + 16i)17-s + (12.5 − 1.97i)18-s + (−14.1 − 24.4i)19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.182 + 0.983i)3-s + (0.249 − 0.433i)4-s + (0.692 − 0.400i)5-s + (−0.235 − 0.666i)6-s + 0.353i·8-s + (−0.933 − 0.359i)9-s + (−0.282 + 0.489i)10-s + (0.779 + 0.449i)11-s + (0.380 + 0.324i)12-s + 0.979·13-s + (0.266 + 0.754i)15-s + (−0.125 − 0.216i)16-s + (1.63 + 0.941i)17-s + (0.698 − 0.109i)18-s + (−0.744 − 1.28i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 - 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.110 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.110 - 0.993i$
Analytic conductor: \(8.01091\)
Root analytic conductor: \(2.83035\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1),\ 0.110 - 0.993i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.989344 + 0.885079i\)
\(L(\frac12)\) \(\approx\) \(0.989344 + 0.885079i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 - 0.707i)T \)
3 \( 1 + (0.548 - 2.94i)T \)
7 \( 1 \)
good5 \( 1 + (-3.46 + 2i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-8.57 - 4.94i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 12.7T + 169T^{2} \)
17 \( 1 + (-27.7 - 16i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (14.1 + 24.4i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-1.22 + 0.707i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 56.5iT - 841T^{2} \)
31 \( 1 + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-25 - 43.3i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 16iT - 1.68e3T^{2} \)
43 \( 1 + 52T + 1.84e3T^{2} \)
47 \( 1 + (-29.4 + 17i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (39.1 + 22.6i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-27.7 - 16i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-16.2 - 28.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-6 + 10.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 89.0iT - 5.04e3T^{2} \)
73 \( 1 + (-16.2 + 28.1i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-20 - 34.6i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 62iT - 6.88e3T^{2} \)
89 \( 1 + (-48.4 + 28i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 24.0T + 9.40e3T^{2} \)
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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

−11.46305314419168625288561830630, −10.58856324452149840586254084489, −9.795750777559129716452290264602, −9.027203834069769156199632146881, −8.312901903910149630260150241743, −6.72469903198228295280651818088, −5.81755711421833705219273368862, −4.83190122784593759813708558423, −3.42429193578936967507687787264, −1.38666176314047816471183184727, 0.966229806536808141994504229512, 2.23214032083372780607422419859, 3.62516040383726778421892496777, 5.81877281437263522418917257787, 6.33624236910576761337338621530, 7.60948964615269136119704423633, 8.371311602185959432037782210316, 9.516832591643050625560755025178, 10.39582310991555718323055724722, 11.45038957973796084912936755705

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