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Properties

Label 2-210-35.13-c1-0-7
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: $\chi_{210} (13, \cdot )$
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

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

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