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Properties

Label 2-210-105.104-c1-0-1
Degree $2$
Conductor $210$
Sign $0.557 - 0.829i$
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 + (2.64 + 2.82i)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 + (0.683 + 0.730i)15-s + 0.250·16-s + 1.15i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.557 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.557 - 0.829i$
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: \((2,\ 210,\ (\ :1/2),\ 0.557 - 0.829i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.526091 + 0.280253i\)
\(L(\frac12)\) \(\approx\) \(0.526091 + 0.280253i\)
\(L(\frac{3}{2})\) 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

−12.42484092115678720597866305895, −11.48640096209861711249102507322, −10.67046101683868007527861514649, −9.510414568825577933244614762903, −8.673767549294495068726572627775, −7.64987263477730370528498934760, −6.22822123708165516814413525698, −5.31438607244306327525566597832, −3.98818470394051712313603532686, −1.50797735151482452778993416098, 0.814560005156893192272222838560, 3.19099826398502361242395411892, 4.83685746274014639830070849504, 6.43895797230488381667563236287, 7.04732132802948817631223870605, 7.992496374551174770801387784704, 9.382343440261723708126784240942, 10.80908245693900382438445152298, 10.95403705842276213026079908409, 11.70785197111325238897137406250

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