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Properties

Label 2-294-7.4-c5-0-9
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 + (51.7 + 89.6i)5-s + 36·6-s − 63.9·8-s + (−40.5 − 70.1i)9-s + (−206. + 358. i)10-s + (−120. + 208. i)11-s + (72 + 124. i)12-s − 805.·13-s + 931.·15-s + (−128 − 221. i)16-s + (−646. + 1.12e3i)17-s + (162 − 280. i)18-s + (−137. − 238. i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.925 + 1.60i)5-s + 0.408·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.654 + 1.13i)10-s + (−0.299 + 0.518i)11-s + (0.144 + 0.249i)12-s − 1.32·13-s + 1.06·15-s + (−0.125 − 0.216i)16-s + (−0.542 + 0.939i)17-s + (0.117 − 0.204i)18-s + (−0.0875 − 0.151i)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

\(L(3)\) \(\approx\) \(1.444589378\)
\(L(\frac12)\) \(\approx\) \(1.444589378\)
\(L(\frac{7}{2})\) 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 \( 1 + (-51.7 - 89.6i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (120. - 208. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 805.T + 3.71e5T^{2} \)
17 \( 1 + (646. - 1.12e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (137. + 238. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (1.89e3 + 3.28e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 1.22e3T + 2.05e7T^{2} \)
31 \( 1 + (-2.81e3 + 4.87e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-4.53e3 - 7.86e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 1.82e4T + 1.15e8T^{2} \)
43 \( 1 + 1.17e4T + 1.47e8T^{2} \)
47 \( 1 + (1.15e4 + 1.99e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (8.83e3 - 1.52e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-9.18e3 + 1.59e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-5.66e3 - 9.80e3i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (1.80e4 - 3.12e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 6.34e4T + 1.80e9T^{2} \)
73 \( 1 + (2.64e4 - 4.58e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-2.42e4 - 4.20e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 1.13e5T + 3.93e9T^{2} \)
89 \( 1 + (-5.41e4 - 9.38e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 9.96e4T + 8.58e9T^{2} \)
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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

−11.50507166395273522856383458769, −10.29671552771805619636232459263, −9.744912345085819228565626336094, −8.305289134725145891190615313530, −7.29433711899035757315862484277, −6.61545259779112006506340454457, −5.86227944470570397242242344409, −4.38179243061646611440402596918, −2.79476834306928668934256448948, −2.15926633740515517148271558999, 0.29408986605787766245766384009, 1.67471991784309015324619986436, 2.81591379628559282954867351180, 4.42972253595802301193796804279, 5.06355014619340073476629186913, 5.94286353236680425702889475061, 7.76098142906066108877135266947, 8.924479093612067559738689628410, 9.512456908400316509229969372022, 10.17936909605998170693083558377

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