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Properties

Label 2-210-7.4-c3-0-12
Degree $2$
Conductor $210$
Sign $-0.602 + 0.797i$
Analytic cond. $12.3904$
Root an. cond. $3.52000$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s + (1.5 − 2.59i)3-s + (−1.99 + 3.46i)4-s + (−2.5 − 4.33i)5-s − 6·6-s + (15.4 + 10.2i)7-s + 7.99·8-s + (−4.5 − 7.79i)9-s + (−5 + 8.66i)10-s + (21.9 − 38.0i)11-s + (6.00 + 10.3i)12-s + 14.2·13-s + (2.21 − 36.9i)14-s − 15.0·15-s + (−8 − 13.8i)16-s + (18.3 − 31.7i)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 + (0.834 + 0.550i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.158 + 0.273i)10-s + (0.601 − 1.04i)11-s + (0.144 + 0.249i)12-s + 0.303·13-s + (0.0423 − 0.705i)14-s − 0.258·15-s + (−0.125 − 0.216i)16-s + (0.261 − 0.452i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.602 + 0.797i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.602 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.602 + 0.797i$
Analytic conductor: \(12.3904\)
Root analytic conductor: \(3.52000\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :3/2),\ -0.602 + 0.797i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.675393 - 1.35651i\)
\(L(\frac12)\) \(\approx\) \(0.675393 - 1.35651i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
3 \( 1 + (-1.5 + 2.59i)T \)
5 \( 1 + (2.5 + 4.33i)T \)
7 \( 1 + (-15.4 - 10.2i)T \)
good11 \( 1 + (-21.9 + 38.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 14.2T + 2.19e3T^{2} \)
17 \( 1 + (-18.3 + 31.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (29.2 + 50.7i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (48.3 + 83.6i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 46.0T + 2.43e4T^{2} \)
31 \( 1 + (-11.6 + 20.1i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (15.6 + 27.0i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 201.T + 6.89e4T^{2} \)
43 \( 1 + 250.T + 7.95e4T^{2} \)
47 \( 1 + (293. + 507. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-165. + 286. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (188. - 327. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (389. + 673. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-90.5 + 156. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 1.11e3T + 3.57e5T^{2} \)
73 \( 1 + (86.8 - 150. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-531. - 920. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 1.42e3T + 5.71e5T^{2} \)
89 \( 1 + (-578. - 1.00e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 487.T + 9.12e5T^{2} \)
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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

−11.60614463443155542120659025918, −10.86819292218383003742447148338, −9.378526235646083987473173691459, −8.570082723507532031684492234357, −7.957897932702868941925685915188, −6.49145874832749720850202151226, −5.07718651364249218533317240116, −3.62644184421679226934622506785, −2.15778279177898211385808547534, −0.74911446209733302330179214928, 1.65664408793833869433363714088, 3.78086682424845259523998542333, 4.75384908206564451705324524583, 6.18670402296274295462315414909, 7.42891375107035685702209861138, 8.104825385850043500498278470573, 9.299038946077012173908479054809, 10.20857799474825881236070524305, 11.03543341060687728098973803679, 12.13926110701556753663108445074

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