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Properties

Label 2-210-105.17-c2-0-11
Degree $2$
Conductor $210$
Sign $0.512 + 0.858i$
Analytic cond. $5.72208$
Root an. cond. $2.39208$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 − 0.366i)2-s + (−2.25 − 1.97i)3-s + (1.73 + i)4-s + (−3.83 + 3.20i)5-s + (2.35 + 3.52i)6-s + (6.92 − 1.00i)7-s + (−1.99 − 2i)8-s + (1.16 + 8.92i)9-s + (6.41 − 2.97i)10-s + (−8.28 − 4.78i)11-s + (−1.92 − 5.68i)12-s + (9.08 + 9.08i)13-s + (−9.83 − 1.16i)14-s + (14.9 + 0.359i)15-s + (1.99 + 3.46i)16-s + (24.4 − 6.55i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.751 − 0.659i)3-s + (0.433 + 0.250i)4-s + (−0.766 + 0.641i)5-s + (0.392 + 0.588i)6-s + (0.989 − 0.143i)7-s + (−0.249 − 0.250i)8-s + (0.129 + 0.991i)9-s + (0.641 − 0.297i)10-s + (−0.753 − 0.434i)11-s + (−0.160 − 0.473i)12-s + (0.698 + 0.698i)13-s + (−0.702 − 0.0832i)14-s + (0.999 + 0.0239i)15-s + (0.124 + 0.216i)16-s + (1.44 − 0.385i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.512 + 0.858i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.512 + 0.858i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.685797 - 0.389209i\)
\(L(\frac12)\) \(\approx\) \(0.685797 - 0.389209i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.366i)T \)
3 \( 1 + (2.25 + 1.97i)T \)
5 \( 1 + (3.83 - 3.20i)T \)
7 \( 1 + (-6.92 + 1.00i)T \)
good11 \( 1 + (8.28 + 4.78i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-9.08 - 9.08i)T + 169iT^{2} \)
17 \( 1 + (-24.4 + 6.55i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (10.4 + 18.1i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-8.88 + 33.1i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 + 6.08T + 841T^{2} \)
31 \( 1 + (-7.22 - 4.17i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-3.80 + 14.2i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 55.3T + 1.68e3T^{2} \)
43 \( 1 + (-57.2 + 57.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (19.9 - 74.4i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-0.717 + 0.192i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-88.7 - 51.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (92.7 - 53.5i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (23.1 - 6.19i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 26.3iT - 5.04e3T^{2} \)
73 \( 1 + (7.55 - 2.02i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-34.5 + 19.9i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-52.9 + 52.9i)T - 6.88e3iT^{2} \)
89 \( 1 + (-24.2 + 14.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-19.4 + 19.4i)T - 9.40e3iT^{2} \)
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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

−11.73994085268118789134050185671, −10.90698346111956035961621120989, −10.59625547874264862526372811729, −8.762092300280489789920784135293, −7.81386090119903220503935548614, −7.16622992795652316295163935793, −5.94601642490504535541416620064, −4.46270434617011264824520038231, −2.58166006363940710942994566308, −0.77884518296075623223439178613, 1.14557951309024550076794781986, 3.70582957544655046618184963538, 5.07822937235254057037972405309, 5.83419554190489500744245092038, 7.65941070450605913542156816476, 8.158842545600484299856343413571, 9.407201827007931919520882653786, 10.43019654146749527232064791988, 11.19216583555742786502188177417, 12.03272892176295652611052034391

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