Properties

Label 2-210-35.27-c1-0-3
Degree $2$
Conductor $210$
Sign $-0.182 - 0.983i$
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 + (0.707 + 0.707i)3-s + 1.00i·4-s + (−1.19 + 1.88i)5-s + 1.00i·6-s + (−2.59 + 0.510i)7-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (−2.18 + 0.489i)10-s + 4.79·11-s + (−0.707 + 0.707i)12-s + (−0.585 − 0.585i)13-s + (−2.19 − 1.47i)14-s + (−2.18 + 0.489i)15-s − 1.00·16-s + (4.10 − 4.10i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.408 + 0.408i)3-s + 0.500i·4-s + (−0.535 + 0.844i)5-s + 0.408i·6-s + (−0.981 + 0.192i)7-s + (−0.250 + 0.250i)8-s + 0.333i·9-s + (−0.689 + 0.154i)10-s + 1.44·11-s + (−0.204 + 0.204i)12-s + (−0.162 − 0.162i)13-s + (−0.587 − 0.394i)14-s + (−0.563 + 0.126i)15-s − 0.250·16-s + (0.995 − 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.182 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.182 - 0.983i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ -0.182 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.980909 + 1.17915i\)
\(L(\frac12)\) \(\approx\) \(0.980909 + 1.17915i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (1.19 - 1.88i)T \)
7 \( 1 + (2.59 - 0.510i)T \)
good11 \( 1 - 4.79T + 11T^{2} \)
13 \( 1 + (0.585 + 0.585i)T + 13iT^{2} \)
17 \( 1 + (-4.10 + 4.10i)T - 17iT^{2} \)
19 \( 1 - 2.36T + 19T^{2} \)
23 \( 1 + (-2.97 + 2.97i)T - 23iT^{2} \)
29 \( 1 - 9.94iT - 29T^{2} \)
31 \( 1 - 3.02iT - 31T^{2} \)
37 \( 1 + (6.10 + 6.10i)T + 37iT^{2} \)
41 \( 1 + 10.9iT - 41T^{2} \)
43 \( 1 + (5.74 - 5.74i)T - 43iT^{2} \)
47 \( 1 + (0.363 - 0.363i)T - 47iT^{2} \)
53 \( 1 + (-2.36 + 2.36i)T - 53iT^{2} \)
59 \( 1 - 2.07T + 59T^{2} \)
61 \( 1 + 5.55iT - 61T^{2} \)
67 \( 1 + (0.979 + 0.979i)T + 67iT^{2} \)
71 \( 1 - 5.25T + 71T^{2} \)
73 \( 1 + (6.11 + 6.11i)T + 73iT^{2} \)
79 \( 1 - 5.10iT - 79T^{2} \)
83 \( 1 + (-3.22 - 3.22i)T + 83iT^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + (8.05 - 8.05i)T - 97iT^{2} \)
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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

−12.56210913115331328650229631776, −11.89673761623021722426624717164, −10.70519030870072895526404885605, −9.578717578577473665403694259245, −8.718700406739651967983221285830, −7.24492340879554747498019042973, −6.71471737668734617668353388232, −5.27930993103168332658868625559, −3.72336725822204180806556407489, −3.06240898261331793995507951865, 1.29239767895444344658424224534, 3.34167161753068869241801123814, 4.19021336581996701433914236049, 5.79873990247017461410395443951, 6.90363486097016942494013969985, 8.170219710318109665533355557263, 9.302859778560181385629316275580, 9.971424071394236465718137641810, 11.67060624783164365014493389285, 12.04693077880495440411083020151

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