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Properties

Label 2-210-35.17-c1-0-3
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: $\chi_{210} (157, \cdot )$
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

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

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