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Properties

Label 2-210-35.12-c1-0-1
Degree $2$
Conductor $210$
Sign $0.740 - 0.671i$
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 + (0.258 − 0.965i)3-s + (0.866 − 0.499i)4-s + (−0.575 + 2.16i)5-s + i·6-s + (1.42 + 2.22i)7-s + (−0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (−0.00342 − 2.23i)10-s + (−0.230 − 0.399i)11-s + (−0.258 − 0.965i)12-s + (4.00 + 4.00i)13-s + (−1.95 − 1.78i)14-s + (1.93 + 1.11i)15-s + (0.500 − 0.866i)16-s + (1.58 + 0.424i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.149 − 0.557i)3-s + (0.433 − 0.249i)4-s + (−0.257 + 0.966i)5-s + 0.408i·6-s + (0.539 + 0.841i)7-s + (−0.249 + 0.249i)8-s + (−0.288 − 0.166i)9-s + (−0.00108 − 0.707i)10-s + (−0.0695 − 0.120i)11-s + (−0.0747 − 0.278i)12-s + (1.11 + 1.11i)13-s + (−0.522 − 0.476i)14-s + (0.500 + 0.287i)15-s + (0.125 − 0.216i)16-s + (0.384 + 0.102i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.740 - 0.671i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.740 - 0.671i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.883658 + 0.341147i\)
\(L(\frac12)\) \(\approx\) \(0.883658 + 0.341147i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (0.575 - 2.16i)T \)
7 \( 1 + (-1.42 - 2.22i)T \)
good11 \( 1 + (0.230 + 0.399i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.00 - 4.00i)T + 13iT^{2} \)
17 \( 1 + (-1.58 - 0.424i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.91 + 5.04i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.14 - 4.26i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 5.53iT - 29T^{2} \)
31 \( 1 + (-0.0280 + 0.0162i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.78 - 2.08i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 10.9iT - 41T^{2} \)
43 \( 1 + (-4.75 + 4.75i)T - 43iT^{2} \)
47 \( 1 + (2.33 + 8.73i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.65 + 0.710i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.958 + 1.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (11.7 + 6.77i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.986 - 3.68i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 8.85T + 71T^{2} \)
73 \( 1 + (1.05 - 3.91i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.38 - 2.53i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.08 + 1.08i)T + 83iT^{2} \)
89 \( 1 + (-5.71 + 9.89i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.51 - 2.51i)T - 97iT^{2} \)
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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

−12.14379233550186805676119256991, −11.42072966693837092449024766930, −10.72330831460769266868579243214, −9.245733584468149038304589239018, −8.559053412533042845789717790749, −7.39096155518597056491359406462, −6.64571643917848465929678344071, −5.44523891706615789002840929009, −3.34846726926555704091763319433, −1.86509201261825377500341035620, 1.15455013650046996411205925818, 3.43686702922714133643860294890, 4.62430861554097338957978269752, 5.94808273490130201285062131742, 7.81200984163599288481469522472, 8.157053290226850765675666047974, 9.344102938784306910167776707712, 10.29629976241174850246589397189, 11.04515967191060836250045929759, 12.11569754985740924455699769614

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