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Properties

Label 2-210-21.2-c2-0-12
Degree $2$
Conductor $210$
Sign $0.732 + 0.681i$
Analytic cond. $5.72208$
Root an. cond. $2.39208$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 0.707i)2-s + (−0.690 + 2.91i)3-s + (0.999 + 1.73i)4-s + (−1.93 − 1.11i)5-s + (2.90 − 3.08i)6-s + (0.399 − 6.98i)7-s − 2.82i·8-s + (−8.04 − 4.03i)9-s + (1.58 + 2.73i)10-s + (0.415 − 0.239i)11-s + (−5.74 + 1.72i)12-s + 8.95·13-s + (−5.43 + 8.27i)14-s + (4.60 − 4.88i)15-s + (−2.00 + 3.46i)16-s + (22.6 − 13.0i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.230 + 0.973i)3-s + (0.249 + 0.433i)4-s + (−0.387 − 0.223i)5-s + (0.484 − 0.514i)6-s + (0.0570 − 0.998i)7-s − 0.353i·8-s + (−0.894 − 0.447i)9-s + (0.158 + 0.273i)10-s + (0.0377 − 0.0218i)11-s + (−0.478 + 0.143i)12-s + 0.688·13-s + (−0.387 + 0.591i)14-s + (0.306 − 0.325i)15-s + (−0.125 + 0.216i)16-s + (1.33 − 0.769i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 + 0.681i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.732 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.732 + 0.681i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.732 + 0.681i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.863104 - 0.339505i\)
\(L(\frac12)\) \(\approx\) \(0.863104 - 0.339505i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.707i)T \)
3 \( 1 + (0.690 - 2.91i)T \)
5 \( 1 + (1.93 + 1.11i)T \)
7 \( 1 + (-0.399 + 6.98i)T \)
good11 \( 1 + (-0.415 + 0.239i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 8.95T + 169T^{2} \)
17 \( 1 + (-22.6 + 13.0i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-1.48 + 2.57i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-27.1 - 15.6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 22.9iT - 841T^{2} \)
31 \( 1 + (19.2 + 33.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-19.3 + 33.5i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 15.5iT - 1.68e3T^{2} \)
43 \( 1 - 11.0T + 1.84e3T^{2} \)
47 \( 1 + (15.0 + 8.68i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-17.5 + 10.1i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-8.37 + 4.83i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-29.9 + 51.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-30.1 - 52.1i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 39.3iT - 5.04e3T^{2} \)
73 \( 1 + (71.7 + 124. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (60.9 - 105. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 108. iT - 6.88e3T^{2} \)
89 \( 1 + (106. + 61.2i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 147.T + 9.40e3T^{2} \)
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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

−11.49003185899410089468145962517, −11.13092190040373445487626180196, −9.998559996988018298156215747277, −9.354318774105696568195718981981, −8.163739555185282983655092019017, −7.17956029467200408779430016546, −5.58992102791409515854776052874, −4.22582091559818998278075371455, −3.26098220509804865939189663970, −0.75983132693810608251243249069, 1.35869682652199775904075730862, 3.03275858547659274434597099673, 5.30561239615368412806077351127, 6.23180538895152868953503171805, 7.23412967411617827897452174621, 8.278484147564290737666279964272, 8.897749817027395383992880901588, 10.41030419571966743466668933320, 11.33156518692773343075315679252, 12.22847536298716120159500273672

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