Properties

Label 2-294-7.4-c3-0-3
Degree $2$
Conductor $294$
Sign $-0.701 + 0.712i$
Analytic cond. $17.3465$
Root an. cond. $4.16492$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)2-s + (−1.5 + 2.59i)3-s + (−1.99 + 3.46i)4-s + (3 + 5.19i)5-s − 6·6-s − 7.99·8-s + (−4.5 − 7.79i)9-s + (−6 + 10.3i)10-s + (−6 + 10.3i)11-s + (−6.00 − 10.3i)12-s − 38·13-s − 18·15-s + (−8 − 13.8i)16-s + (−63 + 109. i)17-s + (9 − 15.5i)18-s + (10 + 17.3i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.268 + 0.464i)5-s − 0.408·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.189 + 0.328i)10-s + (−0.164 + 0.284i)11-s + (−0.144 − 0.249i)12-s − 0.810·13-s − 0.309·15-s + (−0.125 − 0.216i)16-s + (−0.898 + 1.55i)17-s + (0.117 − 0.204i)18-s + (0.120 + 0.209i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.701 + 0.712i$
Analytic conductor: \(17.3465\)
Root analytic conductor: \(4.16492\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :3/2),\ -0.701 + 0.712i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7518815567\)
\(L(\frac12)\) \(\approx\) \(0.7518815567\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 1.73i)T \)
3 \( 1 + (1.5 - 2.59i)T \)
7 \( 1 \)
good5 \( 1 + (-3 - 5.19i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (6 - 10.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 38T + 2.19e3T^{2} \)
17 \( 1 + (63 - 109. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-10 - 17.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (84 + 145. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 30T + 2.43e4T^{2} \)
31 \( 1 + (44 - 76.2i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (127 + 219. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 42T + 6.89e4T^{2} \)
43 \( 1 + 52T + 7.95e4T^{2} \)
47 \( 1 + (48 + 83.1i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (99 - 171. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (330 - 571. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (269 + 465. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (442 - 765. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 792T + 3.57e5T^{2} \)
73 \( 1 + (-109 + 188. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-260 - 450. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 492T + 5.71e5T^{2} \)
89 \( 1 + (-405 - 701. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.15e3T + 9.12e5T^{2} \)
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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

−12.15216547835482556883230330950, −10.74544395123910421163138073958, −10.23664866109193176100804409578, −9.017334210720273120144212115221, −8.043162932754414211602996240800, −6.79539700521427949577129819070, −6.07233260066786283124787533782, −4.87102919833459958809005499664, −3.91458159202108903596718026235, −2.36501203837609088813229901009, 0.24399738006209510674880609782, 1.76082689393026241595696109359, 3.05458293498778730092072136397, 4.72013885712933269044801404359, 5.44477504019580419359726111075, 6.72455575328038498838498487116, 7.78120649299096865921239286355, 9.115861657262346682762043039465, 9.790468505158552916407266786254, 11.05662783862247437044188632211

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