Properties

Label 2-210-35.19-c2-0-5
Degree $2$
Conductor $210$
Sign $0.534 - 0.845i$
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  + (1.22 + 0.707i)2-s + (−0.866 − 1.5i)3-s + (0.999 + 1.73i)4-s + (−4.98 + 0.408i)5-s − 2.44i·6-s + (6.07 + 3.47i)7-s + 2.82i·8-s + (−1.5 + 2.59i)9-s + (−6.39 − 3.02i)10-s + (10.3 + 17.9i)11-s + (1.73 − 2.99i)12-s + 23.2·13-s + (4.99 + 8.54i)14-s + (4.92 + 7.12i)15-s + (−2.00 + 3.46i)16-s + (3.75 + 6.49i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.288 − 0.5i)3-s + (0.249 + 0.433i)4-s + (−0.996 + 0.0817i)5-s − 0.408i·6-s + (0.868 + 0.495i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.639 − 0.302i)10-s + (0.940 + 1.62i)11-s + (0.144 − 0.249i)12-s + 1.78·13-s + (0.356 + 0.610i)14-s + (0.328 + 0.474i)15-s + (−0.125 + 0.216i)16-s + (0.220 + 0.382i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.534 - 0.845i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.534 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.534 - 0.845i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.534 - 0.845i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.64845 + 0.908122i\)
\(L(\frac12)\) \(\approx\) \(1.64845 + 0.908122i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 + (0.866 + 1.5i)T \)
5 \( 1 + (4.98 - 0.408i)T \)
7 \( 1 + (-6.07 - 3.47i)T \)
good11 \( 1 + (-10.3 - 17.9i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 23.2T + 169T^{2} \)
17 \( 1 + (-3.75 - 6.49i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (24.6 + 14.2i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-4.24 - 2.45i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 36.3T + 841T^{2} \)
31 \( 1 + (-1.34 + 0.777i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (8.75 + 5.05i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 3.17iT - 1.68e3T^{2} \)
43 \( 1 - 17.2iT - 1.84e3T^{2} \)
47 \( 1 + (-5.77 + 9.99i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (33.1 - 19.1i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-51.4 + 29.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (26.3 + 15.1i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-103. + 59.4i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 32.6T + 5.04e3T^{2} \)
73 \( 1 + (25.4 + 44.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-68.3 + 118. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 86.0T + 6.88e3T^{2} \)
89 \( 1 + (17.8 + 10.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 17.1T + 9.40e3T^{2} \)
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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

−12.37685504114793475647297969852, −11.45783102713186672331609926488, −10.90488647188239892809339568016, −8.972349998649052553839027386690, −8.103301494410848000706719366622, −7.10308871244479351326941786805, −6.19080037350713985204889250978, −4.75863766667565101054194337068, −3.83303410250370121959289682908, −1.79507797608788025032329963221, 1.04861376606970949969294392109, 3.64660063810606835125942992300, 4.03140926837467652977612265641, 5.52767972688816899090172905101, 6.58207898952117592700174077693, 8.177507396652253660543497648550, 8.852299022143302604075609226276, 10.59182234338609682320246050033, 11.23811059927255017306906724176, 11.58249343783841422822429072182

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