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Properties

Label 2-210-105.89-c1-0-13
Degree $2$
Conductor $210$
Sign $-0.989 + 0.143i$
Analytic cond. $1.67685$
Root an. cond. $1.29493$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 1.65i)3-s + (−0.499 − 0.866i)4-s + (−1.5 + 1.65i)5-s + (1.68 + 0.396i)6-s + (−2.5 − 0.866i)7-s + 0.999·8-s + (−2.5 + 1.65i)9-s + (−0.686 − 2.12i)10-s + (−0.813 + 0.469i)11-s + (−1.18 + 1.26i)12-s − 2·13-s + (2 − 1.73i)14-s + (3.5 + 1.65i)15-s + (−0.5 + 0.866i)16-s + (−5.74 + 3.31i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.288 − 0.957i)3-s + (−0.249 − 0.433i)4-s + (−0.670 + 0.741i)5-s + (0.688 + 0.161i)6-s + (−0.944 − 0.327i)7-s + 0.353·8-s + (−0.833 + 0.552i)9-s + (−0.216 − 0.672i)10-s + (−0.245 + 0.141i)11-s + (−0.342 + 0.364i)12-s − 0.554·13-s + (0.534 − 0.462i)14-s + (0.903 + 0.428i)15-s + (−0.125 + 0.216i)16-s + (−1.39 + 0.804i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.143i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.989 + 0.143i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ -0.989 + 0.143i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.5 + 1.65i)T \)
5 \( 1 + (1.5 - 1.65i)T \)
7 \( 1 + (2.5 + 0.866i)T \)
good11 \( 1 + (0.813 - 0.469i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (5.74 - 3.31i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.686 + 1.18i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.31iT - 29T^{2} \)
31 \( 1 + (-6.55 + 3.78i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.11 - 4.10i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.37T + 41T^{2} \)
43 \( 1 - 1.08iT - 43T^{2} \)
47 \( 1 + (7.37 + 4.25i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.18 - 3.78i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.55 - 11.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (11.0 + 6.38i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.05 - 1.18i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.51iT - 71T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.55 - 7.89i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.8iT - 83T^{2} \)
89 \( 1 + (-0.686 + 1.18i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 15.1T + 97T^{2} \)
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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

−11.91220307748511887340430343588, −10.91456550618470840719725396531, −10.02548187300573671796172917390, −8.591566285069870119259010426548, −7.68250450267913797898968174621, −6.70008915606101660396625301212, −6.26906157516024104207835137048, −4.42826215055449403060272632859, −2.59809620727508189996851675768, 0, 2.89464076198083253935105012705, 4.16883934413190939542220972389, 5.15254909529325554717854524016, 6.69780056937169874837460099749, 8.298034353305992430478819453414, 9.112732912108445086653620975119, 9.839743826847223911230454261298, 10.90905543097664625089540297707, 11.76304922303552471569230619961

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