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Properties

Label 2-210-105.89-c1-0-2
Degree $2$
Conductor $210$
Sign $0.989 - 0.143i$
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 + (−1.68 − 0.396i)3-s + (−0.499 − 0.866i)4-s + (−2.18 − 0.469i)5-s + (1.18 − 1.26i)6-s + (2.5 + 0.866i)7-s + 0.999·8-s + (2.68 + 1.33i)9-s + (1.5 − 1.65i)10-s + (3.68 − 2.12i)11-s + (0.499 + 1.65i)12-s + 2·13-s + (−2 + 1.73i)14-s + (3.5 + 1.65i)15-s + (−0.5 + 0.866i)16-s + (5.74 − 3.31i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.973 − 0.228i)3-s + (−0.249 − 0.433i)4-s + (−0.977 − 0.210i)5-s + (0.484 − 0.515i)6-s + (0.944 + 0.327i)7-s + 0.353·8-s + (0.895 + 0.445i)9-s + (0.474 − 0.524i)10-s + (1.11 − 0.641i)11-s + (0.144 + 0.478i)12-s + 0.554·13-s + (−0.534 + 0.462i)14-s + (0.903 + 0.428i)15-s + (−0.125 + 0.216i)16-s + (1.39 − 0.804i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.989 - 0.143i$
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: \((2,\ 210,\ (\ :1/2),\ 0.989 - 0.143i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.739653 + 0.0535340i\)
\(L(\frac12)\) \(\approx\) \(0.739653 + 0.0535340i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.68 + 0.396i)T \)
5 \( 1 + (2.18 + 0.469i)T \)
7 \( 1 + (-2.5 - 0.866i)T \)
good11 \( 1 + (-3.68 + 2.12i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-5.74 + 3.31i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.18 - 3.78i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.31iT - 29T^{2} \)
31 \( 1 + (2.05 - 1.18i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-10.1 - 5.84i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.62T + 41T^{2} \)
43 \( 1 + 11.0iT - 43T^{2} \)
47 \( 1 + (1.62 + 0.939i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.686 + 1.18i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.05 - 3.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.44 + 1.40i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.55 - 3.78i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.87iT - 71T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.05 + 7.02i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.43iT - 83T^{2} \)
89 \( 1 + (-2.18 + 3.78i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.11T + 97T^{2} \)
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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

−11.88682038130070346399503956965, −11.66979845970239382231437070824, −10.65523159602298609832200275282, −9.256531626824605514185162173315, −8.177342146917017033140003054973, −7.40768360743818186095223884254, −6.20536078412565368655301007716, −5.19349205229589477537241010637, −4.02163189905127661480712420049, −1.08122459295320025299405115064, 1.30910617883881824048118268129, 3.83869120561194407699329693338, 4.49757116710914825427406617272, 6.15564028334702752848442541921, 7.43597084647507459916943322192, 8.321971761656126251589090004627, 9.687150111487389774135513834711, 10.74369495643412998209883849053, 11.24462672998151559793743619503, 12.16628582276577831376366102919

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