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Properties

Label 2-210-105.2-c1-0-4
Degree $2$
Conductor $210$
Sign $-0.853 - 0.521i$
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 + (0.255 + 1.71i)3-s + (−0.866 + 0.499i)4-s + (−2.22 + 0.175i)5-s + (−1.58 + 0.690i)6-s + (2.17 + 1.50i)7-s + (−0.707 − 0.707i)8-s + (−2.86 + 0.875i)9-s + (−0.746 − 2.10i)10-s + (0.149 − 0.0864i)11-s + (−1.07 − 1.35i)12-s + (−2.28 + 2.28i)13-s + (−0.893 + 2.49i)14-s + (−0.869 − 3.77i)15-s + (0.500 − 0.866i)16-s + (3.32 + 0.891i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (0.147 + 0.989i)3-s + (−0.433 + 0.249i)4-s + (−0.996 + 0.0783i)5-s + (−0.648 + 0.281i)6-s + (0.821 + 0.569i)7-s + (−0.249 − 0.249i)8-s + (−0.956 + 0.291i)9-s + (−0.235 − 0.666i)10-s + (0.0451 − 0.0260i)11-s + (−0.311 − 0.391i)12-s + (−0.632 + 0.632i)13-s + (−0.238 + 0.665i)14-s + (−0.224 − 0.974i)15-s + (0.125 − 0.216i)16-s + (0.807 + 0.216i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 - 0.521i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.853 - 0.521i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ -0.853 - 0.521i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.299333 + 1.06428i\)
\(L(\frac12)\) \(\approx\) \(0.299333 + 1.06428i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.255 - 1.71i)T \)
5 \( 1 + (2.22 - 0.175i)T \)
7 \( 1 + (-2.17 - 1.50i)T \)
good11 \( 1 + (-0.149 + 0.0864i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.28 - 2.28i)T - 13iT^{2} \)
17 \( 1 + (-3.32 - 0.891i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.38 + 1.37i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.29 + 1.68i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 0.840T + 29T^{2} \)
31 \( 1 + (-4.06 - 7.04i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.93 + 1.58i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 10.1iT - 41T^{2} \)
43 \( 1 + (3.19 - 3.19i)T - 43iT^{2} \)
47 \( 1 + (0.409 + 1.52i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.73 + 10.2i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (4.86 + 8.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.03 + 10.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.332 - 1.24i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 9.00iT - 71T^{2} \)
73 \( 1 + (-0.0707 - 0.0189i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (12.0 + 6.94i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.33 - 5.33i)T + 83iT^{2} \)
89 \( 1 + (7.02 - 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.33 + 6.33i)T + 97iT^{2} \)
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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

−12.68823427916300692945658670341, −11.69325859488078592084107584477, −10.96641048636832536036007327755, −9.657930864151892557462583760977, −8.595225674976127095001442229833, −7.986995848900130321269479547797, −6.65320666534151068926910212465, −5.08515882587228522567716553045, −4.49173808938666801091863574389, −3.07142914893463519210274193240, 0.963141325563979689047854383126, 2.78972368243263008086018752274, 4.19414768197934277938844714354, 5.49703875449443251111771170030, 7.20267529053958227031100399055, 7.85428085080032064224008461181, 8.807852792869256817427786383085, 10.31032533102020418625026868230, 11.31306055184911664466026273110, 11.96186789557581497057180262517

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