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Properties

Label 2-210-15.14-c2-0-0
Degree $2$
Conductor $210$
Sign $-0.983 + 0.178i$
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  − 1.41·2-s + (−2.94 + 0.554i)3-s + 2.00·4-s + (−0.0300 + 4.99i)5-s + (4.16 − 0.783i)6-s + 2.64i·7-s − 2.82·8-s + (8.38 − 3.26i)9-s + (0.0425 − 7.07i)10-s + 2.58i·11-s + (−5.89 + 1.10i)12-s + 0.180i·13-s − 3.74i·14-s + (−2.68 − 14.7i)15-s + 4.00·16-s − 7.25·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.982 + 0.184i)3-s + 0.500·4-s + (−0.00601 + 0.999i)5-s + (0.694 − 0.130i)6-s + 0.377i·7-s − 0.353·8-s + (0.931 − 0.363i)9-s + (0.00425 − 0.707i)10-s + 0.234i·11-s + (−0.491 + 0.0923i)12-s + 0.0138i·13-s − 0.267i·14-s + (−0.178 − 0.983i)15-s + 0.250·16-s − 0.426·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.178i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.983 + 0.178i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.983 + 0.178i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0195632 - 0.217012i\)
\(L(\frac12)\) \(\approx\) \(0.0195632 - 0.217012i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 + (2.94 - 0.554i)T \)
5 \( 1 + (0.0300 - 4.99i)T \)
7 \( 1 - 2.64iT \)
good11 \( 1 - 2.58iT - 121T^{2} \)
13 \( 1 - 0.180iT - 169T^{2} \)
17 \( 1 + 7.25T + 289T^{2} \)
19 \( 1 + 21.4T + 361T^{2} \)
23 \( 1 + 11.8T + 529T^{2} \)
29 \( 1 + 29.5iT - 841T^{2} \)
31 \( 1 + 49.5T + 961T^{2} \)
37 \( 1 - 43.2iT - 1.36e3T^{2} \)
41 \( 1 + 20.7iT - 1.68e3T^{2} \)
43 \( 1 + 74.2iT - 1.84e3T^{2} \)
47 \( 1 + 7.05T + 2.20e3T^{2} \)
53 \( 1 - 55.0T + 2.80e3T^{2} \)
59 \( 1 - 100. iT - 3.48e3T^{2} \)
61 \( 1 - 8.44T + 3.72e3T^{2} \)
67 \( 1 - 85.7iT - 4.48e3T^{2} \)
71 \( 1 + 97.8iT - 5.04e3T^{2} \)
73 \( 1 + 17.8iT - 5.32e3T^{2} \)
79 \( 1 + 110.T + 6.24e3T^{2} \)
83 \( 1 - 3.08T + 6.88e3T^{2} \)
89 \( 1 - 80.1iT - 7.92e3T^{2} \)
97 \( 1 - 105. iT - 9.40e3T^{2} \)
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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

−12.25215479759267870395283403976, −11.49099518432242501397061370256, −10.62628911221019598228680298809, −10.01785013330839500934086272341, −8.826193603586224547629851451164, −7.43616735952692997504321907261, −6.57535646459938883012171765232, −5.66270743897338975141615197184, −3.99600557666993156629069587256, −2.16069306938527332269288034886, 0.16842808241547641628292569145, 1.67441283774361432738376681434, 4.16150939150219555459447179904, 5.40121722651063696555406402963, 6.48913682578822669882240150439, 7.60825193129420781110862652501, 8.670150493844066719361848912033, 9.688312860955226565061603537996, 10.76085678668643954002833182074, 11.43141505240027367567889160618

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