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Properties

Label 2-210-35.23-c2-0-2
Degree $2$
Conductor $210$
Sign $-0.932 + 0.360i$
Analytic cond. $5.72208$
Root an. cond. $2.39208$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 + 1.36i)2-s + (−0.448 + 1.67i)3-s + (−1.73 + i)4-s + (0.626 + 4.96i)5-s − 2.44·6-s + (−6.84 + 1.45i)7-s + (−2 − 1.99i)8-s + (−2.59 − 1.50i)9-s + (−6.54 + 2.67i)10-s + (−6.59 − 11.4i)11-s + (−0.896 − 3.34i)12-s + (10.7 + 10.7i)13-s + (−4.49 − 8.81i)14-s + (−8.58 − 1.17i)15-s + (1.99 − 3.46i)16-s + (−7.85 − 2.10i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.149 + 0.557i)3-s + (−0.433 + 0.250i)4-s + (0.125 + 0.992i)5-s − 0.408·6-s + (−0.978 + 0.208i)7-s + (−0.250 − 0.249i)8-s + (−0.288 − 0.166i)9-s + (−0.654 + 0.267i)10-s + (−0.599 − 1.03i)11-s + (−0.0747 − 0.278i)12-s + (0.828 + 0.828i)13-s + (−0.321 − 0.629i)14-s + (−0.572 − 0.0783i)15-s + (0.124 − 0.216i)16-s + (−0.462 − 0.123i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 + 0.360i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.932 + 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.932 + 0.360i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.932 + 0.360i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.162545 - 0.871698i\)
\(L(\frac12)\) \(\approx\) \(0.162545 - 0.871698i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 - 1.36i)T \)
3 \( 1 + (0.448 - 1.67i)T \)
5 \( 1 + (-0.626 - 4.96i)T \)
7 \( 1 + (6.84 - 1.45i)T \)
good11 \( 1 + (6.59 + 11.4i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-10.7 - 10.7i)T + 169iT^{2} \)
17 \( 1 + (7.85 + 2.10i)T + (250. + 144.5i)T^{2} \)
19 \( 1 + (-8.35 - 4.82i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (22.6 - 6.06i)T + (458. - 264.5i)T^{2} \)
29 \( 1 - 41.4iT - 841T^{2} \)
31 \( 1 + (18.0 + 31.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-1.25 - 4.68i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 7.31T + 1.68e3T^{2} \)
43 \( 1 + (-48.7 - 48.7i)T + 1.84e3iT^{2} \)
47 \( 1 + (-13.9 - 51.9i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (12.1 - 45.4i)T + (-2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-1.15 + 0.665i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (29.7 - 51.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-117. - 31.5i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 82.6T + 5.04e3T^{2} \)
73 \( 1 + (13.2 - 49.5i)T + (-4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-26.3 - 15.2i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (32.2 + 32.2i)T + 6.88e3iT^{2} \)
89 \( 1 + (-19.1 - 11.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-128. + 128. i)T - 9.40e3iT^{2} \)
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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

−12.91985525653543421861543293042, −11.53535474563643032327166853005, −10.73745669675572710687071113069, −9.670964966604250920961095381882, −8.798362660620232375852819228286, −7.47357881626348010361696358111, −6.28489911469746614010833395620, −5.75508066609392566749657737444, −3.98643713480416751202912696490, −2.97593857603528752309952671321, 0.46246401457790067305604490177, 2.15437769113024771408171408532, 3.80116208915639929157292759504, 5.15783463917767733503600273446, 6.21885805597717125468444744113, 7.62933096284670867178888185019, 8.733747758654906069467434514398, 9.784649741640390852931532578690, 10.58077862882316735057075229391, 11.91766956772041476827249986038

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