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Properties

Label 2-210-15.8-c1-0-6
Degree $2$
Conductor $210$
Sign $0.454 - 0.890i$
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  + (0.707 + 0.707i)2-s + (1.17 + 1.27i)3-s + 1.00i·4-s + (1.37 − 1.76i)5-s + (−0.0727 + 1.73i)6-s + (−0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s + (−0.251 + 2.98i)9-s + (2.21 − 0.275i)10-s − 2.48i·11-s + (−1.27 + 1.17i)12-s + (−1.31 − 1.31i)13-s − 1.00·14-s + (3.86 − 0.314i)15-s − 1.00·16-s + (−2.15 − 2.15i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.676 + 0.736i)3-s + 0.500i·4-s + (0.614 − 0.788i)5-s + (−0.0296 + 0.706i)6-s + (−0.267 + 0.267i)7-s + (−0.250 + 0.250i)8-s + (−0.0838 + 0.996i)9-s + (0.701 − 0.0869i)10-s − 0.750i·11-s + (−0.368 + 0.338i)12-s + (−0.363 − 0.363i)13-s − 0.267·14-s + (0.996 − 0.0812i)15-s − 0.250·16-s + (−0.521 − 0.521i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.454 - 0.890i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.454 - 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61802 + 0.990421i\)
\(L(\frac12)\) \(\approx\) \(1.61802 + 0.990421i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (-1.17 - 1.27i)T \)
5 \( 1 + (-1.37 + 1.76i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good11 \( 1 + 2.48iT - 11T^{2} \)
13 \( 1 + (1.31 + 1.31i)T + 13iT^{2} \)
17 \( 1 + (2.15 + 2.15i)T + 17iT^{2} \)
19 \( 1 - 1.26iT - 19T^{2} \)
23 \( 1 + (3.83 - 3.83i)T - 23iT^{2} \)
29 \( 1 + 2.68T + 29T^{2} \)
31 \( 1 - 10.1T + 31T^{2} \)
37 \( 1 + (-3.73 + 3.73i)T - 37iT^{2} \)
41 \( 1 + 9.92iT - 41T^{2} \)
43 \( 1 + (7.18 + 7.18i)T + 43iT^{2} \)
47 \( 1 + (-9.47 - 9.47i)T + 47iT^{2} \)
53 \( 1 + (6.59 - 6.59i)T - 53iT^{2} \)
59 \( 1 - 1.66T + 59T^{2} \)
61 \( 1 + 1.70T + 61T^{2} \)
67 \( 1 + (4.58 - 4.58i)T - 67iT^{2} \)
71 \( 1 - 2.61iT - 71T^{2} \)
73 \( 1 + (-8.49 - 8.49i)T + 73iT^{2} \)
79 \( 1 + 3.26iT - 79T^{2} \)
83 \( 1 + (-1.42 + 1.42i)T - 83iT^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 + (0.983 - 0.983i)T - 97iT^{2} \)
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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

−12.81499073492998296705146940888, −11.74785818412203747803895146823, −10.38448582324339705097064569475, −9.393474388006192916881574018573, −8.652210122080472282000333363443, −7.66641480091777796723891579616, −6.02220617038204122398743570422, −5.16309261596472506497321238424, −3.98799416265247684181864274874, −2.55555609396432981797362454262, 1.93212281243424322520034456760, 2.98172642142741850343871635492, 4.43678162369875853557329985430, 6.26523180759230104449077602376, 6.83538800703120159445223131827, 8.136285632938063642619404614609, 9.549219820013578151926881517134, 10.14576597075755573368396466041, 11.40697819411479665375789830229, 12.36799365191247373262675325582

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