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Properties

Label 2-210-105.89-c1-0-6
Degree $2$
Conductor $210$
Sign $0.680 - 0.732i$
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  + (−0.5 + 0.866i)2-s + (1.72 − 0.178i)3-s + (−0.499 − 0.866i)4-s + (0.792 + 2.09i)5-s + (−0.707 + 1.58i)6-s + (1.41 − 2.23i)7-s + 0.999·8-s + (2.93 − 0.614i)9-s + (−2.20 − 0.358i)10-s + (−4.05 + 2.34i)11-s + (−1.01 − 1.40i)12-s + 1.04·13-s + (1.22 + 2.34i)14-s + (1.73 + 3.46i)15-s + (−0.5 + 0.866i)16-s + (2.73 − 1.58i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.994 − 0.102i)3-s + (−0.249 − 0.433i)4-s + (0.354 + 0.935i)5-s + (−0.288 + 0.645i)6-s + (0.534 − 0.845i)7-s + 0.353·8-s + (0.978 − 0.204i)9-s + (−0.697 − 0.113i)10-s + (−1.22 + 0.706i)11-s + (−0.293 − 0.404i)12-s + 0.289·13-s + (0.328 + 0.626i)14-s + (0.448 + 0.893i)15-s + (−0.125 + 0.216i)16-s + (0.664 − 0.383i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.680 - 0.732i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.680 - 0.732i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31028 + 0.571084i\)
\(L(\frac12)\) \(\approx\) \(1.31028 + 0.571084i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-1.72 + 0.178i)T \)
5 \( 1 + (-0.792 - 2.09i)T \)
7 \( 1 + (-1.41 + 2.23i)T \)
good11 \( 1 + (4.05 - 2.34i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.04T + 13T^{2} \)
17 \( 1 + (-2.73 + 1.58i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.23 + 0.715i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.23 - 3.87i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.92iT - 29T^{2} \)
31 \( 1 + (5.73 - 3.31i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.30 - 1.33i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.04T + 41T^{2} \)
43 \( 1 + 6.92iT - 43T^{2} \)
47 \( 1 + (9.71 + 5.60i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.5 + 4.33i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.28 + 9.15i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3 - 1.73i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-12.3 + 7.13i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.92iT - 71T^{2} \)
73 \( 1 + (-1.75 - 3.03i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.73 - 9.93i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.06iT - 83T^{2} \)
89 \( 1 + (2.45 - 4.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
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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

−12.91450187092821377301485544887, −11.23244719735388242579958967474, −10.16183185390146069809510085364, −9.717412492057569000655887908228, −8.176262829699036216214073135600, −7.56490328684589288289174177570, −6.76416240195911724411607307932, −5.17934693072363739182501772432, −3.65663298282855415352594429622, −2.05832466972383533892055145329, 1.73942389558075121942672037980, 3.02337625024810887376488043207, 4.58347159173169057815828921323, 5.75611632274706211980508366703, 7.86155200026443182175449680163, 8.402577606365663566728841656355, 9.157330701291988757636565271580, 10.15895374887549632896971506751, 11.15423696537857143911594240482, 12.57920708631186809332591734756

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