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Properties

Label 2-294-7.4-c5-0-23
Degree $2$
Conductor $294$
Sign $-0.701 + 0.712i$
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 + (−33 − 57.1i)5-s + 36·6-s + 63.9·8-s + (−40.5 − 70.1i)9-s + (−132 + 228. i)10-s + (30 − 51.9i)11-s + (−72 − 124. i)12-s + 658·13-s + 594·15-s + (−128 − 221. i)16-s + (−207 + 358. i)17-s + (−162 + 280. i)18-s + (478 + 827. i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.590 − 1.02i)5-s + 0.408·6-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.417 + 0.722i)10-s + (0.0747 − 0.129i)11-s + (−0.144 − 0.249i)12-s + 1.07·13-s + 0.681·15-s + (−0.125 − 0.216i)16-s + (−0.173 + 0.300i)17-s + (−0.117 + 0.204i)18-s + (0.303 + 0.526i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.701 + 0.712i$
Analytic conductor: \(47.1528\)
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: \((2,\ 294,\ (\ :5/2),\ -0.701 + 0.712i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.9040649862\)
\(L(\frac12)\) \(\approx\) \(0.9040649862\)
\(L(\frac{7}{2})\) 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 \( 1 + (33 + 57.1i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-30 + 51.9i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 658T + 3.71e5T^{2} \)
17 \( 1 + (207 - 358. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-478 - 827. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (300 + 519. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 5.57e3T + 2.05e7T^{2} \)
31 \( 1 + (1.79e3 - 3.11e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-4.22e3 - 7.32e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.91e4T + 1.15e8T^{2} \)
43 \( 1 - 1.33e4T + 1.47e8T^{2} \)
47 \( 1 + (9.84e3 + 1.70e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-1.56e4 + 2.70e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-1.31e4 + 2.28e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.55e4 + 2.69e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-8.40e3 + 1.45e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 6.12e3T + 1.80e9T^{2} \)
73 \( 1 + (1.27e4 - 2.21e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (3.72e4 + 6.44e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 6.46e3T + 3.93e9T^{2} \)
89 \( 1 + (1.63e4 + 2.83e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.66e5T + 8.58e9T^{2} \)
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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

−10.59666994623695620131299422592, −9.731704632076195621139454986724, −8.544790368079452624790885755570, −8.291098781389758607023202823617, −6.61750436039157874416055478279, −5.26319697609641260632932560471, −4.26530410902136749831566192152, −3.33930705836991288869032007897, −1.46689882851051516437205575695, −0.35789569649946744305125083339, 1.07341417662103448072030332220, 2.77355139327966991472389943108, 4.16073735690507415198346542558, 5.64158934229286571932637687758, 6.61929038410843121444332952987, 7.28652246979139996360830080716, 8.187325767977830632337476066303, 9.241316422756062759757480390361, 10.51185963827847038168948863488, 11.16405690785750962517957729520

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