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Properties

Label 2-210-105.23-c1-0-14
Degree $2$
Conductor $210$
Sign $0.202 + 0.979i$
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  + (0.965 − 0.258i)2-s + (0.517 − 1.65i)3-s + (0.866 − 0.499i)4-s + (−2.16 + 0.564i)5-s + (0.0717 − 1.73i)6-s + (0.113 − 2.64i)7-s + (0.707 − 0.707i)8-s + (−2.46 − 1.70i)9-s + (−1.94 + 1.10i)10-s + (3.54 − 2.04i)11-s + (−0.378 − 1.69i)12-s + (3.69 + 3.69i)13-s + (−0.574 − 2.58i)14-s + (−0.186 + 3.86i)15-s + (0.500 − 0.866i)16-s + (−1.93 + 7.20i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.298 − 0.954i)3-s + (0.433 − 0.249i)4-s + (−0.967 + 0.252i)5-s + (0.0292 − 0.706i)6-s + (0.0427 − 0.999i)7-s + (0.249 − 0.249i)8-s + (−0.821 − 0.569i)9-s + (−0.614 + 0.349i)10-s + (1.06 − 0.616i)11-s + (−0.109 − 0.487i)12-s + (1.02 + 1.02i)13-s + (−0.153 − 0.690i)14-s + (−0.0482 + 0.998i)15-s + (0.125 − 0.216i)16-s + (−0.468 + 1.74i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.202 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.202 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.202 + 0.979i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.202 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32025 - 1.07482i\)
\(L(\frac12)\) \(\approx\) \(1.32025 - 1.07482i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 + (-0.517 + 1.65i)T \)
5 \( 1 + (2.16 - 0.564i)T \)
7 \( 1 + (-0.113 + 2.64i)T \)
good11 \( 1 + (-3.54 + 2.04i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.69 - 3.69i)T + 13iT^{2} \)
17 \( 1 + (1.93 - 7.20i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.12 + 1.22i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.284 - 1.06i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 1.48T + 29T^{2} \)
31 \( 1 + (-2.62 - 4.54i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.43 - 5.34i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 6.74iT - 41T^{2} \)
43 \( 1 + (-1.27 - 1.27i)T + 43iT^{2} \)
47 \( 1 + (-2.97 + 0.796i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (5.50 + 1.47i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.09 + 8.81i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.814 - 1.41i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.39 + 0.909i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 5.89iT - 71T^{2} \)
73 \( 1 + (-1.44 + 5.38i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-8.18 - 4.72i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.31 - 6.31i)T - 83iT^{2} \)
89 \( 1 + (-6.71 + 11.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.21 - 5.21i)T - 97iT^{2} \)
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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

−12.21604427871169883003824723752, −11.31464385766723634156891726366, −10.76610371762161268394221002481, −8.891510588438113763704168470989, −8.045523131576668329823429202590, −6.75891221387893470482870481919, −6.35373083195546001500014369109, −4.16586975346754349561748443119, −3.51403891862938503392300347998, −1.44721842335910059659827791069, 2.83534913932069269751524373324, 4.02460289122564030229382458215, 4.94227832077998079515260800840, 6.13581395439066485204465819812, 7.64696603872572342404595995836, 8.673281926294124993831394272678, 9.453349844906686044358084938152, 10.95809479148683119660392992046, 11.63900787671392049017181168897, 12.46758464190536042422796704732

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