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Properties

Label 2-210-35.19-c2-0-11
Degree $2$
Conductor $210$
Sign $0.974 + 0.225i$
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 + (4.99 + 0.0121i)5-s − 2.44i·6-s + (0.686 − 6.96i)7-s + 2.82i·8-s + (−1.5 + 2.59i)9-s + (6.11 + 3.55i)10-s + (−2.20 − 3.82i)11-s + (1.73 − 2.99i)12-s + 19.3·13-s + (5.76 − 8.04i)14-s + (−4.31 − 7.51i)15-s + (−2.00 + 3.46i)16-s + (−1.48 − 2.57i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.288 − 0.5i)3-s + (0.249 + 0.433i)4-s + (0.999 + 0.00243i)5-s − 0.408i·6-s + (0.0980 − 0.995i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.611 + 0.355i)10-s + (−0.200 − 0.347i)11-s + (0.144 − 0.249i)12-s + 1.49·13-s + (0.411 − 0.574i)14-s + (−0.287 − 0.500i)15-s + (−0.125 + 0.216i)16-s + (−0.0872 − 0.151i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.225i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.974 + 0.225i$
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: \((2,\ 210,\ (\ :1),\ 0.974 + 0.225i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.32716 - 0.265688i\)
\(L(\frac12)\) \(\approx\) \(2.32716 - 0.265688i\)
\(L(2)\) 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 \( 1 + (-4.99 - 0.0121i)T \)
7 \( 1 + (-0.686 + 6.96i)T \)
good11 \( 1 + (2.20 + 3.82i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 19.3T + 169T^{2} \)
17 \( 1 + (1.48 + 2.57i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-2.06 - 1.19i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (1.29 + 0.748i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 39.2T + 841T^{2} \)
31 \( 1 + (35.0 - 20.2i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (36.8 + 21.2i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 57.4iT - 1.68e3T^{2} \)
43 \( 1 + 53.4iT - 1.84e3T^{2} \)
47 \( 1 + (38.5 - 66.8i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (81.3 - 46.9i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (8.24 - 4.75i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (35.0 + 20.2i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (53.1 - 30.6i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 46.1T + 5.04e3T^{2} \)
73 \( 1 + (1.76 + 3.06i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-47.5 + 82.4i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 94.2T + 6.88e3T^{2} \)
89 \( 1 + (-133. - 76.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 122.T + 9.40e3T^{2} \)
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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

−12.38843679966849913382612056667, −11.05704938134068806488687817548, −10.49310845715749895794702129519, −9.004425059148338669716437761142, −7.86965258501876419214129425626, −6.68865701137411859458595651700, −6.00514464327507917144778444654, −4.80866427319067818211161236953, −3.28066945355850318056040215532, −1.41119451324348145536189083293, 1.82135127744399885415905101593, 3.28934610903465516905829911784, 4.84715854160747799933675590765, 5.75543566389884370015180879034, 6.53102027674399047909270634011, 8.490030587206112601060720258942, 9.396686149712211198945098831294, 10.37881638648109030362412444962, 11.19931597752100373258849766649, 12.20925336010892969765061983577

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