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Properties

Label 2-210-15.2-c1-0-3
Degree $2$
Conductor $210$
Sign $0.452 - 0.891i$
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 + (1.53 + 0.799i)3-s − 1.00i·4-s + (1.91 + 1.15i)5-s + (−1.65 + 0.521i)6-s + (−0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + (1.72 + 2.45i)9-s + (−2.17 + 0.536i)10-s − 1.70i·11-s + (0.799 − 1.53i)12-s + (0.921 − 0.921i)13-s + 1.00·14-s + (2.01 + 3.30i)15-s − 1.00·16-s + (−4.76 + 4.76i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.887 + 0.461i)3-s − 0.500i·4-s + (0.856 + 0.516i)5-s + (−0.674 + 0.212i)6-s + (−0.267 − 0.267i)7-s + (0.250 + 0.250i)8-s + (0.574 + 0.818i)9-s + (−0.686 + 0.169i)10-s − 0.514i·11-s + (0.230 − 0.443i)12-s + (0.255 − 0.255i)13-s + 0.267·14-s + (0.521 + 0.853i)15-s − 0.250·16-s + (−1.15 + 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.452 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.452 - 0.891i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.452 - 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15067 + 0.706817i\)
\(L(\frac12)\) \(\approx\) \(1.15067 + 0.706817i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.53 - 0.799i)T \)
5 \( 1 + (-1.91 - 1.15i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good11 \( 1 + 1.70iT - 11T^{2} \)
13 \( 1 + (-0.921 + 0.921i)T - 13iT^{2} \)
17 \( 1 + (4.76 - 4.76i)T - 17iT^{2} \)
19 \( 1 + 5.94iT - 19T^{2} \)
23 \( 1 + (-2.49 - 2.49i)T + 23iT^{2} \)
29 \( 1 + 5.19T + 29T^{2} \)
31 \( 1 + 3.40T + 31T^{2} \)
37 \( 1 + (1.02 + 1.02i)T + 37iT^{2} \)
41 \( 1 + 10.9iT - 41T^{2} \)
43 \( 1 + (-8.17 + 8.17i)T - 43iT^{2} \)
47 \( 1 + (0.436 - 0.436i)T - 47iT^{2} \)
53 \( 1 + (6.87 + 6.87i)T + 53iT^{2} \)
59 \( 1 - 0.686T + 59T^{2} \)
61 \( 1 + 1.74T + 61T^{2} \)
67 \( 1 + (-1.03 - 1.03i)T + 67iT^{2} \)
71 \( 1 + 12.2iT - 71T^{2} \)
73 \( 1 + (4.59 - 4.59i)T - 73iT^{2} \)
79 \( 1 + 7.19iT - 79T^{2} \)
83 \( 1 + (-6.64 - 6.64i)T + 83iT^{2} \)
89 \( 1 + 2.25T + 89T^{2} \)
97 \( 1 + (-13.2 - 13.2i)T + 97iT^{2} \)
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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

−13.09171361585204936954554944649, −10.89658506766293438990180908140, −10.60668212662487319034628541748, −9.251480583288586362056131439642, −8.905571582732146123354658275966, −7.52852794330711657789198832209, −6.56875199841125508474468000975, −5.32302502875250458482126368217, −3.68395181293659637421381507639, −2.16968843195902093665338916735, 1.63528317652275521628122922433, 2.80495176298863783918874763525, 4.44846747768548499038172437545, 6.17812817438296348338348541126, 7.32764906626788083898280276593, 8.499462262735726569601232634309, 9.320956233722039855581863996782, 9.833247771025517871283872178328, 11.22703821656383962338882224134, 12.47760213791630138235055062534

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