Properties

Label 2-210-35.17-c1-0-4
Degree $2$
Conductor $210$
Sign $0.998 + 0.0581i$
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 + (0.965 − 0.258i)3-s + (−0.866 − 0.499i)4-s + (−0.619 − 2.14i)5-s + i·6-s + (2.25 − 1.38i)7-s + (0.707 − 0.707i)8-s + (0.866 − 0.499i)9-s + (2.23 − 0.0421i)10-s + (0.582 − 1.00i)11-s + (−0.965 − 0.258i)12-s + (1.92 + 1.92i)13-s + (0.756 + 2.53i)14-s + (−1.15 − 1.91i)15-s + (0.500 + 0.866i)16-s + (−0.00560 − 0.0209i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (0.557 − 0.149i)3-s + (−0.433 − 0.249i)4-s + (−0.276 − 0.960i)5-s + 0.408i·6-s + (0.851 − 0.524i)7-s + (0.249 − 0.249i)8-s + (0.288 − 0.166i)9-s + (0.706 − 0.0133i)10-s + (0.175 − 0.304i)11-s + (−0.278 − 0.0747i)12-s + (0.533 + 0.533i)13-s + (0.202 + 0.677i)14-s + (−0.298 − 0.494i)15-s + (0.125 + 0.216i)16-s + (−0.00136 − 0.00507i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.998 + 0.0581i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.998 + 0.0581i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29824 - 0.0377713i\)
\(L(\frac12)\) \(\approx\) \(1.29824 - 0.0377713i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (0.619 + 2.14i)T \)
7 \( 1 + (-2.25 + 1.38i)T \)
good11 \( 1 + (-0.582 + 1.00i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.92 - 1.92i)T + 13iT^{2} \)
17 \( 1 + (0.00560 + 0.0209i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.989 - 1.71i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.93 + 1.85i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 5.60iT - 29T^{2} \)
31 \( 1 + (-6.86 - 3.96i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.74 + 10.2i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 2.48iT - 41T^{2} \)
43 \( 1 + (7.87 - 7.87i)T - 43iT^{2} \)
47 \( 1 + (3.94 + 1.05i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.757 + 2.82i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.34 - 9.25i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.15 - 1.82i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (14.0 - 3.76i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.51T + 71T^{2} \)
73 \( 1 + (-3.61 + 0.969i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-1.39 + 0.805i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.74 - 9.74i)T + 83iT^{2} \)
89 \( 1 + (1.80 + 3.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.265 + 0.265i)T - 97iT^{2} \)
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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

−12.47608013516470047706097581433, −11.47962457585013741333819625533, −10.20429495321258319031558490204, −9.029757771205715518741685533407, −8.300779372724878730211070339901, −7.60595890527782043477981838268, −6.25884517732142933645903793986, −4.86792573586020518090768216252, −3.89503077960273755160118998234, −1.41645034528243912336708911340, 2.08617977862729191280577807170, 3.31749176113465260657155649326, 4.57332368747268293430299394846, 6.19310007891483326097294582256, 7.75746621560027527834761700521, 8.316328983104400920052097011401, 9.642995674837636508506406357051, 10.41195691573617313580216375271, 11.48567347225861494605423847321, 12.01173688154935877773576918808

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