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Properties

Label 2-210-35.17-c1-0-7
Degree $2$
Conductor $210$
Sign $-0.972 - 0.232i$
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 + (−2.21 − 0.296i)5-s + i·6-s + (−1.87 − 1.86i)7-s + (−0.707 + 0.707i)8-s + (0.866 − 0.499i)9-s + (−0.860 + 2.06i)10-s + (−2.74 + 4.75i)11-s + (0.965 + 0.258i)12-s + (−2.41 − 2.41i)13-s + (−2.28 + 1.33i)14-s + (2.21 − 0.286i)15-s + (0.500 + 0.866i)16-s + (−0.548 − 2.04i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.557 + 0.149i)3-s + (−0.433 − 0.249i)4-s + (−0.991 − 0.132i)5-s + 0.408i·6-s + (−0.709 − 0.704i)7-s + (−0.249 + 0.249i)8-s + (0.288 − 0.166i)9-s + (−0.272 + 0.652i)10-s + (−0.827 + 1.43i)11-s + (0.278 + 0.0747i)12-s + (−0.670 − 0.670i)13-s + (−0.611 + 0.355i)14-s + (0.572 − 0.0740i)15-s + (0.125 + 0.216i)16-s + (−0.132 − 0.496i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.232i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 - 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.972 - 0.232i$
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: \((2,\ 210,\ (\ :1/2),\ -0.972 - 0.232i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0328125 + 0.278700i\)
\(L(\frac12)\) \(\approx\) \(0.0328125 + 0.278700i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 + 0.296i)T \)
7 \( 1 + (1.87 + 1.86i)T \)
good11 \( 1 + (2.74 - 4.75i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.41 + 2.41i)T + 13iT^{2} \)
17 \( 1 + (0.548 + 2.04i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.49 + 6.05i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.69 - 0.454i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 0.684iT - 29T^{2} \)
31 \( 1 + (-4.82 - 2.78i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.53 + 9.46i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 2.50iT - 41T^{2} \)
43 \( 1 + (1.95 - 1.95i)T - 43iT^{2} \)
47 \( 1 + (3.40 + 0.912i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.43 + 9.08i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.08 - 8.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.01 + 0.585i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.61 - 2.57i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 + (-4.70 + 1.26i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-7.21 + 4.16i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.05 + 4.05i)T + 83iT^{2} \)
89 \( 1 + (3.59 + 6.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.1 - 13.1i)T - 97iT^{2} \)
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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

−11.91640524750993738293377166332, −10.83257286345553837499839029018, −10.21828635282317829358136837790, −9.177646326591376917687427519327, −7.62892555407222454778862662086, −6.85519020053812598030555980323, −5.02094548734674292231612110782, −4.32542690307275920512639207640, −2.82375918523490761074377208946, −0.23162469866311737990408210640, 3.11204942711360329858358162761, 4.50885825426416710056657530088, 5.85536499091866613338858774614, 6.57033491024264016155187399592, 7.909588318937060328364389026538, 8.569633462695173900299504971745, 9.989660184804894886195247257862, 11.12886218248996157589705320408, 12.06887215345752848377416507072, 12.77044983265811661638634353653

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