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Properties

Label 2-210-7.2-c5-0-1
Degree $2$
Conductor $210$
Sign $-0.926 + 0.375i$
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 + (−40.9 − 122. i)7-s + 63.9·8-s + (−40.5 + 70.1i)9-s + (50 + 86.6i)10-s + (52.9 + 91.6i)11-s + (72 − 124. i)12-s + 416.·13-s + (508. + 103. i)14-s + 225·15-s + (−128 + 221. i)16-s + (−904. − 1.56e3i)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 + (−0.316 − 0.948i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.158 + 0.273i)10-s + (0.131 + 0.228i)11-s + (0.144 − 0.249i)12-s + 0.682·13-s + (0.692 + 0.141i)14-s + 0.258·15-s + (−0.125 + 0.216i)16-s + (−0.758 − 1.31i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 + 0.375i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.926 + 0.375i$
Analytic conductor: \(33.6806\)
Root analytic conductor: \(5.80349\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :5/2),\ -0.926 + 0.375i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.2396338743\)
\(L(\frac12)\) \(\approx\) \(0.2396338743\)
\(L(\frac{7}{2})\) 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 \( 1 + (40.9 + 122. i)T \)
good11 \( 1 + (-52.9 - 91.6i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 416.T + 3.71e5T^{2} \)
17 \( 1 + (904. + 1.56e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (967. - 1.67e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (2.42e3 - 4.19e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 2.79e3T + 2.05e7T^{2} \)
31 \( 1 + (-2.28e3 - 3.95e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (348. - 603. i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 8.15e3T + 1.15e8T^{2} \)
43 \( 1 + 1.08e4T + 1.47e8T^{2} \)
47 \( 1 + (-2.32e3 + 4.02e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-1.59e4 - 2.76e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (1.48e4 + 2.57e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-1.50e3 + 2.61e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (5.53e3 + 9.59e3i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 7.20e4T + 1.80e9T^{2} \)
73 \( 1 + (3.34e4 + 5.79e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-2.84e4 + 4.92e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 5.54e3T + 3.93e9T^{2} \)
89 \( 1 + (5.83e4 - 1.01e5i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 3.45e4T + 8.58e9T^{2} \)
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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

−11.95882732615349237827468531384, −10.74805803937258525378013334673, −9.885745304625613490490563955057, −9.140009371022647451783970890598, −8.071510539981663954862104962731, −7.07393256102878243325528115893, −5.90545023036167703583265188994, −4.64872164112814007966150500519, −3.57320132135709584020358518128, −1.55852815308144100889312999304, 0.07475782184884673660145682690, 1.85482996052398075882624582190, 2.74543852939687236527382275007, 4.12679678188131471212762581947, 5.96575358871704060355586465922, 6.75233887447594351671998781496, 8.407222653614307502558691759690, 8.737422648539940787721649661191, 10.01634237232745905062143890974, 10.97924167669362768409270027834

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