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Properties

Label 2-210-35.9-c1-0-5
Degree $2$
Conductor $210$
Sign $-0.881 + 0.471i$
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  + (−0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (−1.40 + 1.74i)5-s + 0.999·6-s + (−1.45 − 2.20i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (2.08 − 0.806i)10-s + (−3.05 − 5.28i)11-s + (−0.866 − 0.499i)12-s − 1.68i·13-s + (0.155 + 2.64i)14-s + (0.344 − 2.20i)15-s + (−0.5 + 0.866i)16-s + (−5.92 + 3.41i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.627 + 0.778i)5-s + 0.408·6-s + (−0.550 − 0.835i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (0.659 − 0.255i)10-s + (−0.920 − 1.59i)11-s + (−0.249 − 0.144i)12-s − 0.468i·13-s + (0.0416 + 0.705i)14-s + (0.0888 − 0.570i)15-s + (−0.125 + 0.216i)16-s + (−1.43 + 0.829i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.881 + 0.471i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ -0.881 + 0.471i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0488374 - 0.194776i\)
\(L(\frac12)\) \(\approx\) \(0.0488374 - 0.194776i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (1.40 - 1.74i)T \)
7 \( 1 + (1.45 + 2.20i)T \)
good11 \( 1 + (3.05 + 5.28i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.68iT - 13T^{2} \)
17 \( 1 + (5.92 - 3.41i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.844 + 1.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.00 - 1.15i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.41T + 29T^{2} \)
31 \( 1 + (-0.344 - 0.596i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.00 + 1.15i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.14T + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 + (-2.65 - 1.53i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-9.05 + 5.23i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.52 + 6.09i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.73 - 8.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.5 - 6.10i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (3.46 - 2i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.65 + 6.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.5iT - 83T^{2} \)
89 \( 1 + (6.10 - 10.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.95iT - 97T^{2} \)
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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

−11.47427374014625008174812329540, −10.82039851351661696210212353712, −10.46830176904107950588088826543, −9.071225537670682139442312906904, −7.940921997257087855278060063713, −6.97018989415019580340131792242, −5.85911548491535680522049894881, −4.01737986502264136138026553278, −2.99549560776680708503235728872, −0.20784542283062059267330616768, 2.18094162019118144585178366430, 4.54322169588407932818350892413, 5.51435256257384286195598399349, 6.90317961158238638032577605431, 7.64755630673902033372345778661, 8.937131549420374951514936444879, 9.566567520377297182741022386545, 10.87097615146053844906457509092, 11.90038244706145381156114660752, 12.59245921585225520848873814647

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