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Properties

Label 2-210-35.27-c1-0-1
Degree $2$
Conductor $210$
Sign $0.255 - 0.966i$
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  + (0.707 + 0.707i)2-s + (−0.707 − 0.707i)3-s + 1.00i·4-s + (−0.489 + 2.18i)5-s − 1.00i·6-s + (2.19 + 1.47i)7-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (−1.88 + 1.19i)10-s + 0.0296·11-s + (0.707 − 0.707i)12-s + (0.585 + 0.585i)13-s + (0.510 + 2.59i)14-s + (1.88 − 1.19i)15-s − 1.00·16-s + (−1.72 + 1.72i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (−0.218 + 0.975i)5-s − 0.408i·6-s + (0.830 + 0.557i)7-s + (−0.250 + 0.250i)8-s + 0.333i·9-s + (−0.597 + 0.378i)10-s + 0.00893·11-s + (0.204 − 0.204i)12-s + (0.162 + 0.162i)13-s + (0.136 + 0.693i)14-s + (0.487 − 0.308i)15-s − 0.250·16-s + (−0.417 + 0.417i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.255 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.255 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.255 - 0.966i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.255 - 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09577 + 0.843798i\)
\(L(\frac12)\) \(\approx\) \(1.09577 + 0.843798i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (0.489 - 2.18i)T \)
7 \( 1 + (-2.19 - 1.47i)T \)
good11 \( 1 - 0.0296T + 11T^{2} \)
13 \( 1 + (-0.585 - 0.585i)T + 13iT^{2} \)
17 \( 1 + (1.72 - 1.72i)T - 17iT^{2} \)
19 \( 1 - 5.77T + 19T^{2} \)
23 \( 1 + (0.393 - 0.393i)T - 23iT^{2} \)
29 \( 1 + 9.70iT - 29T^{2} \)
31 \( 1 + 6.39iT - 31T^{2} \)
37 \( 1 + (3.72 + 3.72i)T + 37iT^{2} \)
41 \( 1 + 0.514iT - 41T^{2} \)
43 \( 1 + (-7.16 + 7.16i)T - 43iT^{2} \)
47 \( 1 + (7.77 - 7.77i)T - 47iT^{2} \)
53 \( 1 + (5.77 - 5.77i)T - 53iT^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 + 10.7iT - 61T^{2} \)
67 \( 1 + (-2.39 - 2.39i)T + 67iT^{2} \)
71 \( 1 - 6.64T + 71T^{2} \)
73 \( 1 + (-5.12 - 5.12i)T + 73iT^{2} \)
79 \( 1 - 9.86iT - 79T^{2} \)
83 \( 1 + (-0.150 - 0.150i)T + 83iT^{2} \)
89 \( 1 + 4.38T + 89T^{2} \)
97 \( 1 + (2.47 - 2.47i)T - 97iT^{2} \)
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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

−12.52137339138236942626903738539, −11.54779140023239422447101632567, −11.10699428758309960885042764660, −9.640268409796265393962433072013, −8.161880762063185220705712211547, −7.44832810903025104752822674096, −6.32632872406074717113135047630, −5.44466296063345922593282241770, −4.01936466230030139506797869458, −2.36612723289257541217669530095, 1.26063777627821884652102145149, 3.52601338123622890353398598238, 4.80341502719104702526969016835, 5.29970524877266752999990670496, 6.98534057028152754912842965379, 8.316081009693176530943168556787, 9.364225906699848837239782985028, 10.44548257492555592109486190094, 11.35733154106299528687904963845, 12.03221334801179817175762330647

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