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Properties

Label 2-210-105.104-c1-0-4
Degree $2$
Conductor $210$
Sign $0.883 - 0.468i$
Analytic cond. $1.67685$
Root an. cond. $1.29493$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + (−1.68 − 0.420i)3-s + 4-s + (1.08 + 1.95i)5-s + (−1.68 − 0.420i)6-s + (0.595 + 2.57i)7-s + 8-s + (2.64 + 1.41i)9-s + (1.08 + 1.95i)10-s − 2.82i·11-s + (−1.68 − 0.420i)12-s + 3.36·13-s + (0.595 + 2.57i)14-s + (−1 − 3.74i)15-s + 16-s − 4.75i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.970 − 0.242i)3-s + 0.5·4-s + (0.485 + 0.874i)5-s + (−0.685 − 0.171i)6-s + (0.224 + 0.974i)7-s + 0.353·8-s + (0.881 + 0.471i)9-s + (0.343 + 0.618i)10-s − 0.852i·11-s + (−0.485 − 0.121i)12-s + 0.931·13-s + (0.159 + 0.688i)14-s + (−0.258 − 0.966i)15-s + 0.250·16-s − 1.15i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.883 - 0.468i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.883 - 0.468i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45970 + 0.363453i\)
\(L(\frac12)\) \(\approx\) \(1.45970 + 0.363453i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (1.68 + 0.420i)T \)
5 \( 1 + (-1.08 - 1.95i)T \)
7 \( 1 + (-0.595 - 2.57i)T \)
good11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 - 3.36T + 13T^{2} \)
17 \( 1 + 4.75iT - 17T^{2} \)
19 \( 1 - 5.59iT - 19T^{2} \)
23 \( 1 + 7.29T + 23T^{2} \)
29 \( 1 - 0.500iT - 29T^{2} \)
31 \( 1 + 3.06iT - 31T^{2} \)
37 \( 1 + 3.32iT - 37T^{2} \)
41 \( 1 + 4.33T + 41T^{2} \)
43 \( 1 + 10.3iT - 43T^{2} \)
47 \( 1 + 7.82iT - 47T^{2} \)
53 \( 1 - 8.58T + 53T^{2} \)
59 \( 1 - 2.16T + 59T^{2} \)
61 \( 1 + 2.52iT - 61T^{2} \)
67 \( 1 - 10.3iT - 67T^{2} \)
71 \( 1 + 9.81iT - 71T^{2} \)
73 \( 1 - 5.53T + 73T^{2} \)
79 \( 1 - 3.29T + 79T^{2} \)
83 \( 1 - 6.97iT - 83T^{2} \)
89 \( 1 + 15.8T + 89T^{2} \)
97 \( 1 + 14.6T + 97T^{2} \)
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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

−12.21569496258664607393584433417, −11.63323282010065916909796753037, −10.81131762723626001337174884613, −9.857982235892728360473628902166, −8.276463861135320021997353473805, −6.96254946178784730539025155305, −5.88099972654355601767906937151, −5.55109237085821038101068658167, −3.72393751990918871139912241278, −2.09773636479614506714933654805, 1.46644578542261602795297905447, 4.05085180131574216898327624018, 4.73279922876582492350683131456, 5.91866758129159635228733639978, 6.81519823234917862555710953708, 8.183019380567621115488490942066, 9.690257343409448732607526137265, 10.50621479055246978482605209831, 11.40469293514864609771462746373, 12.43337890134681571774601774094

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