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Properties

Label 2-294-7.2-c5-0-33
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: $\chi_{294} (79, \cdot )$
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

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

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