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Properties

Label 2-210-35.33-c1-0-2
Degree $2$
Conductor $210$
Sign $0.326 - 0.945i$
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  + (0.258 + 0.965i)2-s + (−0.965 − 0.258i)3-s + (−0.866 + 0.499i)4-s + (2.14 − 0.637i)5-s − i·6-s + (0.153 + 2.64i)7-s + (−0.707 − 0.707i)8-s + (0.866 + 0.499i)9-s + (1.17 + 1.90i)10-s + (2.27 + 3.94i)11-s + (0.965 − 0.258i)12-s + (1.77 − 1.77i)13-s + (−2.51 + 0.831i)14-s + (−2.23 + 0.0614i)15-s + (0.500 − 0.866i)16-s + (−1.06 + 3.98i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.557 − 0.149i)3-s + (−0.433 + 0.249i)4-s + (0.958 − 0.285i)5-s − 0.408i·6-s + (0.0579 + 0.998i)7-s + (−0.249 − 0.249i)8-s + (0.288 + 0.166i)9-s + (0.370 + 0.602i)10-s + (0.686 + 1.18i)11-s + (0.278 − 0.0747i)12-s + (0.493 − 0.493i)13-s + (−0.671 + 0.222i)14-s + (−0.577 + 0.0158i)15-s + (0.125 − 0.216i)16-s + (−0.258 + 0.966i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.326 - 0.945i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.326 - 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02072 + 0.726943i\)
\(L(\frac12)\) \(\approx\) \(1.02072 + 0.726943i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 + (-2.14 + 0.637i)T \)
7 \( 1 + (-0.153 - 2.64i)T \)
good11 \( 1 + (-2.27 - 3.94i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.77 + 1.77i)T - 13iT^{2} \)
17 \( 1 + (1.06 - 3.98i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.88 + 3.27i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.77 - 2.08i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 1.55iT - 29T^{2} \)
31 \( 1 + (-3.37 + 1.94i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.95 + 11.0i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 11.3iT - 41T^{2} \)
43 \( 1 + (-0.367 - 0.367i)T + 43iT^{2} \)
47 \( 1 + (-4.87 + 1.30i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.18 - 8.14i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.221 + 0.383i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.09 - 4.09i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.99 + 2.41i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 6.68T + 71T^{2} \)
73 \( 1 + (4.20 + 1.12i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.08 + 2.35i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.21 - 3.21i)T - 83iT^{2} \)
89 \( 1 + (-3.02 + 5.23i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.462 - 0.462i)T + 97iT^{2} \)
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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

−12.54013383755049740465024312630, −11.96679936971645756803391702569, −10.49090301574992147502720874762, −9.471995945025126883541663939374, −8.654961941552564583871764429002, −7.30197388710283977123716378039, −6.08396863049308863697829727986, −5.59137904283572491786727749428, −4.26495674560421212646514405992, −2.01706198445780068345730315920, 1.32224703976757942174941374817, 3.29339107792868393892518739025, 4.55649662089767325610627652906, 5.92733094626113377187008057205, 6.71151834370965900777277595147, 8.392033743801759010220753942268, 9.674337320744718434090851402351, 10.27928374677627225067248685451, 11.25840790421550905374176119380, 11.89069780751085147592149972271

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