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Properties

Label 2-210-7.4-c3-0-14
Degree $2$
Conductor $210$
Sign $-0.997 + 0.0767i$
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  + (−1 − 1.73i)2-s + (1.5 − 2.59i)3-s + (−1.99 + 3.46i)4-s + (2.5 + 4.33i)5-s − 6·6-s + (−2.58 − 18.3i)7-s + 7.99·8-s + (−4.5 − 7.79i)9-s + (5 − 8.66i)10-s + (−3.91 + 6.77i)11-s + (6.00 + 10.3i)12-s − 32.8·13-s + (−29.1 + 22.8i)14-s + 15.0·15-s + (−8 − 13.8i)16-s + (34.4 − 59.6i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s − 0.408·6-s + (−0.139 − 0.990i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.158 − 0.273i)10-s + (−0.107 + 0.185i)11-s + (0.144 + 0.249i)12-s − 0.700·13-s + (−0.556 + 0.435i)14-s + 0.258·15-s + (−0.125 − 0.216i)16-s + (0.491 − 0.851i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0767i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.997 + 0.0767i$
Analytic conductor: \(12.3904\)
Root analytic conductor: \(3.52000\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :3/2),\ -0.997 + 0.0767i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0357321 - 0.929938i\)
\(L(\frac12)\) \(\approx\) \(0.0357321 - 0.929938i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
3 \( 1 + (-1.5 + 2.59i)T \)
5 \( 1 + (-2.5 - 4.33i)T \)
7 \( 1 + (2.58 + 18.3i)T \)
good11 \( 1 + (3.91 - 6.77i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 32.8T + 2.19e3T^{2} \)
17 \( 1 + (-34.4 + 59.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (22.8 + 39.5i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (108. + 188. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 126.T + 2.43e4T^{2} \)
31 \( 1 + (84.5 - 146. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-121. - 210. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 286.T + 6.89e4T^{2} \)
43 \( 1 + 54.2T + 7.95e4T^{2} \)
47 \( 1 + (113. + 196. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-26.8 + 46.5i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-435. + 753. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-232. - 402. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-105. + 182. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 236.T + 3.57e5T^{2} \)
73 \( 1 + (-176. + 305. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (391. + 677. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 956.T + 5.71e5T^{2} \)
89 \( 1 + (93.0 + 161. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 10.1T + 9.12e5T^{2} \)
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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

−11.41998221344457437223856641601, −10.33656575364361046338514559006, −9.733965007949919295577958511199, −8.444757614506127608378020084009, −7.40550340846519955359493379448, −6.61916243599681186708255808427, −4.79634137783997384358518690071, −3.36025165098572157238674815081, −2.10339035321407649469355062346, −0.40822855465790036132351985741, 1.99826675736112997550102117907, 3.79477233171524835756394076682, 5.33966141220376589050668163248, 5.95771697167499847451237349937, 7.60156812747363580014502578183, 8.447297496274576150933458403723, 9.443802148732140131768285673507, 9.998192771849248882535002298754, 11.36591621180435090092839046234, 12.44926197338647358025334461894

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