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Properties

Label 2-210-105.104-c1-0-0
Degree $2$
Conductor $210$
Sign $-0.934 - 0.355i$
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 + (0.420 + 1.68i)3-s + 4-s + (−1.95 + 1.08i)5-s + (−0.420 − 1.68i)6-s + (−2.37 + 1.16i)7-s − 8-s + (−2.64 + 1.41i)9-s + (1.95 − 1.08i)10-s − 2.82i·11-s + (0.420 + 1.68i)12-s − 0.841·13-s + (2.37 − 1.16i)14-s + (−2.64 − 2.82i)15-s + 16-s + 1.19i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.242 + 0.970i)3-s + 0.5·4-s + (−0.874 + 0.485i)5-s + (−0.171 − 0.685i)6-s + (−0.898 + 0.439i)7-s − 0.353·8-s + (−0.881 + 0.471i)9-s + (0.618 − 0.343i)10-s − 0.852i·11-s + (0.121 + 0.485i)12-s − 0.233·13-s + (0.635 − 0.311i)14-s + (−0.683 − 0.730i)15-s + 0.250·16-s + 0.288i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.355i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.934 - 0.355i$
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: \((2,\ 210,\ (\ :1/2),\ -0.934 - 0.355i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0876499 + 0.477162i\)
\(L(\frac12)\) \(\approx\) \(0.0876499 + 0.477162i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.420 - 1.68i)T \)
5 \( 1 + (1.95 - 1.08i)T \)
7 \( 1 + (2.37 - 1.16i)T \)
good11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 + 0.841T + 13T^{2} \)
17 \( 1 - 1.19iT - 17T^{2} \)
19 \( 1 - 4.55iT - 19T^{2} \)
23 \( 1 + 3.29T + 23T^{2} \)
29 \( 1 - 7.98iT - 29T^{2} \)
31 \( 1 - 5.53iT - 31T^{2} \)
37 \( 1 - 10.8iT - 37T^{2} \)
41 \( 1 - 7.82T + 41T^{2} \)
43 \( 1 + 4.65iT - 43T^{2} \)
47 \( 1 + 4.33iT - 47T^{2} \)
53 \( 1 - 12.5T + 53T^{2} \)
59 \( 1 + 3.91T + 59T^{2} \)
61 \( 1 + 10.0iT - 61T^{2} \)
67 \( 1 - 4.65iT - 67T^{2} \)
71 \( 1 - 12.6iT - 71T^{2} \)
73 \( 1 - 3.06T + 73T^{2} \)
79 \( 1 + 7.29T + 79T^{2} \)
83 \( 1 - 7.70iT - 83T^{2} \)
89 \( 1 + 12.8T + 89T^{2} \)
97 \( 1 - 8.11T + 97T^{2} \)
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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

−12.46981942696961717509699283412, −11.60673368756340125306596684827, −10.59914020653738493442102167560, −9.982751956312556026890550486476, −8.796401453902772342643355396169, −8.167834709241382308367639119446, −6.79235097908172184367863594354, −5.59034028837580024410420999398, −3.80786978680849867275871315589, −2.92475385206497316195899961231, 0.48365635517016648534400069353, 2.51723553906575536474786011466, 4.14657155283287519386544003128, 6.05417945385030425722634806431, 7.29115884899780409439796462339, 7.65801762965544907626892232906, 8.956531284431379058380447728014, 9.705005802204891788305208645998, 11.11066877573098231187879158297, 12.02420267042255358636181873377

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