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Properties

Label 2-294-7.2-c5-0-7
Degree $2$
Conductor $294$
Sign $0.605 - 0.795i$
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 + (13 − 22.5i)5-s + 36·6-s − 63.9·8-s + (−40.5 + 70.1i)9-s + (−51.9 − 90.0i)10-s + (179 + 310. i)11-s + (72 − 124. i)12-s − 332·13-s + 234·15-s + (−128 + 221. i)16-s + (63 + 109. i)17-s + (162 + 280. i)18-s + (−1.10e3 + 1.90e3i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.232 − 0.402i)5-s + 0.408·6-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.164 − 0.284i)10-s + (0.446 + 0.772i)11-s + (0.144 − 0.249i)12-s − 0.544·13-s + 0.268·15-s + (−0.125 + 0.216i)16-s + (0.0528 + 0.0915i)17-s + (0.117 + 0.204i)18-s + (−0.699 + 1.21i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.605 - 0.795i$
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: \((2,\ 294,\ (\ :5/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.136566781\)
\(L(\frac12)\) \(\approx\) \(2.136566781\)
\(L(\frac{7}{2})\) 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 \( 1 + (-13 + 22.5i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-179 - 310. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 332T + 3.71e5T^{2} \)
17 \( 1 + (-63 - 109. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (1.10e3 - 1.90e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-1.07e3 + 1.85e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 3.61e3T + 2.05e7T^{2} \)
31 \( 1 + (-2.83e3 - 4.90e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-1.46e3 + 2.53e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 2.14e3T + 1.15e8T^{2} \)
43 \( 1 - 6.38e3T + 1.47e8T^{2} \)
47 \( 1 + (3.26e3 - 5.64e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-5.35e3 - 9.26e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-2.12e4 - 3.68e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (2.24e4 - 3.88e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-724 - 1.25e3i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 4.40e3T + 1.80e9T^{2} \)
73 \( 1 + (-1.02e4 - 1.77e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (3.26e4 - 5.64e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 1.02e5T + 3.93e9T^{2} \)
89 \( 1 + (6.40e4 - 1.10e5i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 1.13e5T + 8.58e9T^{2} \)
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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

−10.95972840037414925701694915414, −10.15188181225150597881321490842, −9.364826522892778695543790551364, −8.513954658983610688932399918090, −7.17846294992893375536625963588, −5.81949269244628833384782371554, −4.74644636969917032823404774823, −3.90387884159142652007025271696, −2.54445364563080335554005424394, −1.34688145532103827685278266608, 0.49602306900848312024214238818, 2.29781881329715037036176194058, 3.43198829009702849543240627499, 4.79561554778738951173206647394, 6.05793948831282596266349838456, 6.79804518967372025373211861673, 7.73713937743768409212497700691, 8.748365335055392703734936576431, 9.607192464078556703127653208561, 10.97247486440045530362241191083

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