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Properties

Label 2-210-105.104-c1-0-2
Degree $2$
Conductor $210$
Sign $0.292 - 0.956i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.292 - 0.956i$
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: \((2,\ 210,\ (\ :1/2),\ 0.292 - 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.783954 + 0.580152i\)
\(L(\frac12)\) \(\approx\) \(0.783954 + 0.580152i\)
\(L(\frac{3}{2})\) 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

−12.23317997274146408236705262688, −11.24613646534519062677635366966, −10.32516534965258085023166223418, −9.950417911632992110487277096821, −8.799014898163811355587185031272, −7.68347720387636679831921238752, −6.36390616775611896111975643618, −5.27955350781227689943737883212, −3.87001045777439211895733533279, −2.00707429774840263502805584996, 1.23833068979534546307491174632, 2.54850379386885800302515650190, 5.14364607542033502650341569336, 6.03062248722615198177416766796, 7.14937987269971100614848093073, 8.457458289503195629925774513476, 8.807040075492804539693129489767, 10.25596171137238654144571220648, 11.34210440329987769084435234083, 11.95433998778976005217219653187

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