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Properties

Label 2-210-7.4-c5-0-5
Degree $2$
Conductor $210$
Sign $0.262 - 0.965i$
Analytic cond. $33.6806$
Root an. cond. $5.80349$
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 + (12.5 + 21.6i)5-s − 36·6-s + (−127. − 25.9i)7-s + 63.9·8-s + (−40.5 − 70.1i)9-s + (50 − 86.6i)10-s + (188. − 325. i)11-s + (72 + 124. i)12-s − 444.·13-s + (163. + 491. i)14-s + 225·15-s + (−128 − 221. i)16-s + (435. − 753. i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s − 0.408·6-s + (−0.979 − 0.200i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.158 − 0.273i)10-s + (0.468 − 0.811i)11-s + (0.144 + 0.249i)12-s − 0.728·13-s + (0.223 + 0.670i)14-s + 0.258·15-s + (−0.125 − 0.216i)16-s + (0.365 − 0.632i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 - 0.965i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.262 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.262 - 0.965i$
Analytic conductor: \(33.6806\)
Root analytic conductor: \(5.80349\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :5/2),\ 0.262 - 0.965i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.4973153373\)
\(L(\frac12)\) \(\approx\) \(0.4973153373\)
\(L(\frac{7}{2})\) 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 \( 1 + (-4.5 + 7.79i)T \)
5 \( 1 + (-12.5 - 21.6i)T \)
7 \( 1 + (127. + 25.9i)T \)
good11 \( 1 + (-188. + 325. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 444.T + 3.71e5T^{2} \)
17 \( 1 + (-435. + 753. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (131. + 228. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-2.11e3 - 3.65e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 7.29e3T + 2.05e7T^{2} \)
31 \( 1 + (4.35e3 - 7.53e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-1.38e3 - 2.39e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 1.06e4T + 1.15e8T^{2} \)
43 \( 1 + 1.58e4T + 1.47e8T^{2} \)
47 \( 1 + (579. + 1.00e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (1.61e4 - 2.79e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.82e4 - 3.16e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-2.07e4 - 3.59e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-1.50e4 + 2.60e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 3.76e4T + 1.80e9T^{2} \)
73 \( 1 + (-1.61e4 + 2.80e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-3.23e4 - 5.60e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 1.05e5T + 3.93e9T^{2} \)
89 \( 1 + (-1.11e4 - 1.93e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 3.72e4T + 8.58e9T^{2} \)
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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

−11.65368694092620670983782791348, −10.76031374472297198882012577333, −9.560183608293954880455573728721, −9.084384276469910987815855444251, −7.59550978786611421825868479288, −6.84384781519450086954504104914, −5.51387837433213202674313189511, −3.60027544279768995254150874579, −2.81629656119590376384862427592, −1.27731307688945001351646522983, 0.16693286097785041014254497589, 2.11563153641465900457639889221, 3.79443352287129430532508538398, 5.01311689096808007965365737469, 6.17109246052970509643285717742, 7.22677546892351465845304094055, 8.403830648638118308327261598286, 9.551503142255729944648925065429, 9.733996025884143704383647974913, 11.06233013749502221068447907902

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