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Properties

Label 2-210-35.17-c1-0-2
Degree $2$
Conductor $210$
Sign $0.781 + 0.624i$
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 + (1.58 + 1.57i)5-s − i·6-s + (2.22 + 1.42i)7-s + (−0.707 + 0.707i)8-s + (0.866 − 0.499i)9-s + (1.93 − 1.12i)10-s + (−0.230 + 0.399i)11-s + (−0.965 − 0.258i)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 + (0.424 + 1.58i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.557 − 0.149i)3-s + (−0.433 − 0.249i)4-s + (0.708 + 0.706i)5-s − 0.408i·6-s + (0.841 + 0.539i)7-s + (−0.249 + 0.249i)8-s + (0.288 − 0.166i)9-s + (0.611 − 0.354i)10-s + (−0.0695 + 0.120i)11-s + (−0.278 − 0.0747i)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 + (0.102 + 0.384i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 + 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.781 + 0.624i$
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: \((2,\ 210,\ (\ :1/2),\ 0.781 + 0.624i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56705 - 0.548914i\)
\(L(\frac12)\) \(\approx\) \(1.56705 - 0.548914i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.58 - 1.57i)T \)
7 \( 1 + (-2.22 - 1.42i)T \)
good11 \( 1 + (0.230 - 0.399i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.00 + 4.00i)T + 13iT^{2} \)
17 \( 1 + (-0.424 - 1.58i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.91 + 5.04i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.26 + 1.14i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 5.53iT - 29T^{2} \)
31 \( 1 + (-0.0280 - 0.0162i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.08 + 7.78i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 10.9iT - 41T^{2} \)
43 \( 1 + (-4.75 + 4.75i)T - 43iT^{2} \)
47 \( 1 + (8.73 + 2.33i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.710 - 2.65i)T + (-45.8 + 26.5i)T^{2} \)
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 \( 1 + (-3.68 + 0.986i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 8.85T + 71T^{2} \)
73 \( 1 + (3.91 - 1.05i)T + (63.2 - 36.5i)T^{2} \)
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

−12.41828274456425037966830983137, −11.19346876092171409259866604123, −10.38901531546462174394773256614, −9.484148648362775033103842281831, −8.413182528099728054795354239828, −7.30680121991679317671086659358, −5.85174457770002452087657130262, −4.70581778769026170633451698121, −2.95674623297793804247756243829, −2.04979528530124723330652741844, 1.98316127493371393164939762444, 4.14301755081032027935863666418, 4.95986499915299502521852283522, 6.26739036211060459455843179103, 7.59589174763944484998321177458, 8.349600339300158844651392579721, 9.447089293593112907159615284474, 10.16516156558236767835243482359, 11.71695878423568027536590445651, 12.65825349430546447250109237317

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