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Properties

Label 2-294-7.4-c5-0-0
Degree $2$
Conductor $294$
Sign $-0.947 + 0.318i$
Analytic cond. $47.1528$
Root an. cond. $6.86679$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 3.46i)2-s + (−4.5 + 7.79i)3-s + (−7.99 + 13.8i)4-s + (23.4 + 40.6i)5-s + 36·6-s + 63.9·8-s + (−40.5 − 70.1i)9-s + (93.8 − 162. i)10-s + (43.7 − 75.7i)11-s + (−72 − 124. i)12-s − 754.·13-s − 422.·15-s + (−128 − 221. i)16-s + (−724. + 1.25e3i)17-s + (−162 + 280. i)18-s + (1.27e3 + 2.19e3i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.419 + 0.727i)5-s + 0.408·6-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.296 − 0.514i)10-s + (0.108 − 0.188i)11-s + (−0.144 − 0.249i)12-s − 1.23·13-s − 0.484·15-s + (−0.125 − 0.216i)16-s + (−0.608 + 1.05i)17-s + (−0.117 + 0.204i)18-s + (0.807 + 1.39i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.947 + 0.318i$
Analytic conductor: \(47.1528\)
Root analytic conductor: \(6.86679\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :5/2),\ -0.947 + 0.318i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.09673113957\)
\(L(\frac12)\) \(\approx\) \(0.09673113957\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 + 3.46i)T \)
3 \( 1 + (4.5 - 7.79i)T \)
7 \( 1 \)
good5 \( 1 + (-23.4 - 40.6i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-43.7 + 75.7i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 754.T + 3.71e5T^{2} \)
17 \( 1 + (724. - 1.25e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-1.27e3 - 2.19e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-456. - 790. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 173.T + 2.05e7T^{2} \)
31 \( 1 + (-2.26e3 + 3.92e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (3.41e3 + 5.91e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.30e4T + 1.15e8T^{2} \)
43 \( 1 + 1.22e4T + 1.47e8T^{2} \)
47 \( 1 + (-6.74e3 - 1.16e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-4.83e3 + 8.38e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-1.51e4 + 2.63e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-366. - 634. i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.31e4 - 4.01e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 3.29e3T + 1.80e9T^{2} \)
73 \( 1 + (-5.11e3 + 8.86e3i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (4.92e4 + 8.53e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 8.77e4T + 3.93e9T^{2} \)
89 \( 1 + (3.45e4 + 5.98e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 4.21e4T + 8.58e9T^{2} \)
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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

−11.35742294697540137856217390569, −10.16543587832941026705486569023, −10.09011316311036178499398654197, −8.836845556545799678054839145430, −7.70727137067394487127927241462, −6.54066614978360558478846212700, −5.42042055800965840966283142916, −4.10646509164222393857938209105, −2.98171201421556822787595819215, −1.74815761770332222277528006705, 0.03226152419804914656898436522, 1.18334164418394292871357046483, 2.62665226951817819904782040170, 4.85252359248570732967000159265, 5.21880537331044452674625487182, 6.77333228664798831982057537074, 7.23359244430428747218196624152, 8.554253892381539553403876427170, 9.305623520191681893769455143523, 10.16617148361565866131316119028

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