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Properties

Label 2-210-5.4-c3-0-15
Degree $2$
Conductor $210$
Sign $-0.617 + 0.786i$
Analytic cond. $12.3904$
Root an. cond. $3.52000$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s − 3i·3-s − 4·4-s + (8.79 + 6.89i)5-s − 6·6-s − 7i·7-s + 8i·8-s − 9·9-s + (13.7 − 17.5i)10-s + 25.5·11-s + 12i·12-s − 89.1i·13-s − 14·14-s + (20.6 − 26.3i)15-s + 16·16-s − 18.7i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (0.786 + 0.617i)5-s − 0.408·6-s − 0.377i·7-s + 0.353i·8-s − 0.333·9-s + (0.436 − 0.556i)10-s + 0.701·11-s + 0.288i·12-s − 1.90i·13-s − 0.267·14-s + (0.356 − 0.454i)15-s + 0.250·16-s − 0.268i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.617 + 0.786i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.617 + 0.786i$
Analytic conductor: \(12.3904\)
Root analytic conductor: \(3.52000\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :3/2),\ -0.617 + 0.786i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.766001 - 1.57408i\)
\(L(\frac12)\) \(\approx\) \(0.766001 - 1.57408i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
3 \( 1 + 3iT \)
5 \( 1 + (-8.79 - 6.89i)T \)
7 \( 1 + 7iT \)
good11 \( 1 - 25.5T + 1.33e3T^{2} \)
13 \( 1 + 89.1iT - 2.19e3T^{2} \)
17 \( 1 + 18.7iT - 4.91e3T^{2} \)
19 \( 1 - 11.7T + 6.85e3T^{2} \)
23 \( 1 + 159. iT - 1.21e4T^{2} \)
29 \( 1 + 19.7T + 2.43e4T^{2} \)
31 \( 1 + 126.T + 2.97e4T^{2} \)
37 \( 1 + 282. iT - 5.06e4T^{2} \)
41 \( 1 + 144.T + 6.89e4T^{2} \)
43 \( 1 - 326. iT - 7.95e4T^{2} \)
47 \( 1 - 117. iT - 1.03e5T^{2} \)
53 \( 1 + 634. iT - 1.48e5T^{2} \)
59 \( 1 - 515.T + 2.05e5T^{2} \)
61 \( 1 - 395.T + 2.26e5T^{2} \)
67 \( 1 - 842. iT - 3.00e5T^{2} \)
71 \( 1 - 402.T + 3.57e5T^{2} \)
73 \( 1 + 303. iT - 3.89e5T^{2} \)
79 \( 1 + 266.T + 4.93e5T^{2} \)
83 \( 1 + 753. iT - 5.71e5T^{2} \)
89 \( 1 - 842.T + 7.04e5T^{2} \)
97 \( 1 - 1.20e3iT - 9.12e5T^{2} \)
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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

−11.48573050285837304076095393710, −10.59845725327994864353904631754, −9.891216338465306160358270996927, −8.683191839071403868529133662633, −7.50987924174870906328609297680, −6.36452384848430569024430398647, −5.26276454794895947959319157823, −3.48644613800062838769438118577, −2.33240538198941657152457753491, −0.78274227703291363125779283803, 1.71959400382534641412037710552, 3.87389408965135190523375148902, 4.98250944213868018471642099689, 5.98592879194487554795898484134, 6.99844601033642910552607900615, 8.584882977934690859713367202028, 9.235759071472101273230784678825, 9.870489203997161843207295125271, 11.38531306186047716965721096961, 12.24677085197691069933292317533

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