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Properties

Label 2-210-105.83-c2-0-15
Degree $2$
Conductor $210$
Sign $-0.596 - 0.802i$
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 + i)2-s + (0.947 + 2.84i)3-s + 2i·4-s + (4.64 + 1.84i)5-s + (−1.89 + 3.79i)6-s + (−6.25 + 3.13i)7-s + (−2 + 2i)8-s + (−7.20 + 5.39i)9-s + (2.79 + 6.49i)10-s − 2.08i·11-s + (−5.69 + 1.89i)12-s + (8.39 − 8.39i)13-s + (−9.39 − 3.12i)14-s + (−0.854 + 14.9i)15-s − 4·16-s + (4.96 − 4.96i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (0.315 + 0.948i)3-s + 0.5i·4-s + (0.929 + 0.369i)5-s + (−0.316 + 0.632i)6-s + (−0.893 + 0.448i)7-s + (−0.250 + 0.250i)8-s + (−0.800 + 0.599i)9-s + (0.279 + 0.649i)10-s − 0.189i·11-s + (−0.474 + 0.157i)12-s + (0.645 − 0.645i)13-s + (−0.671 − 0.222i)14-s + (−0.0569 + 0.998i)15-s − 0.250·16-s + (0.292 − 0.292i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.596 - 0.802i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.596 - 0.802i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.596 - 0.802i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.00649 + 2.00303i\)
\(L(\frac12)\) \(\approx\) \(1.00649 + 2.00303i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 + (-0.947 - 2.84i)T \)
5 \( 1 + (-4.64 - 1.84i)T \)
7 \( 1 + (6.25 - 3.13i)T \)
good11 \( 1 + 2.08iT - 121T^{2} \)
13 \( 1 + (-8.39 + 8.39i)T - 169iT^{2} \)
17 \( 1 + (-4.96 + 4.96i)T - 289iT^{2} \)
19 \( 1 - 17.3T + 361T^{2} \)
23 \( 1 + (3.08 - 3.08i)T - 529iT^{2} \)
29 \( 1 + 39.1T + 841T^{2} \)
31 \( 1 - 42.3iT - 961T^{2} \)
37 \( 1 + (-36.7 + 36.7i)T - 1.36e3iT^{2} \)
41 \( 1 - 15.5T + 1.68e3T^{2} \)
43 \( 1 + (-22.8 - 22.8i)T + 1.84e3iT^{2} \)
47 \( 1 + (-33.4 + 33.4i)T - 2.20e3iT^{2} \)
53 \( 1 + (-59.7 + 59.7i)T - 2.80e3iT^{2} \)
59 \( 1 + 48.9iT - 3.48e3T^{2} \)
61 \( 1 - 82.9iT - 3.72e3T^{2} \)
67 \( 1 + (54.8 - 54.8i)T - 4.48e3iT^{2} \)
71 \( 1 - 74.9iT - 5.04e3T^{2} \)
73 \( 1 + (-75.1 + 75.1i)T - 5.32e3iT^{2} \)
79 \( 1 - 3.61iT - 6.24e3T^{2} \)
83 \( 1 + (103. + 103. i)T + 6.88e3iT^{2} \)
89 \( 1 + 24.4iT - 7.92e3T^{2} \)
97 \( 1 + (35.3 + 35.3i)T + 9.40e3iT^{2} \)
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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

−12.81375682505648843907296532935, −11.43999544138477039365760872428, −10.36698840966710295365496932006, −9.505178964676561283243015003891, −8.721330970635145102384605772951, −7.29174369560188543347751799549, −5.89340020542541934027771848918, −5.40691795744999710021241460638, −3.66921823500557824232008791983, −2.74404710966424375294900753714, 1.14133172483907125353069157791, 2.52378713945907049046085331946, 3.92281298751125272435384776460, 5.70342216697372104691479374792, 6.41437892383259325885980938675, 7.60900854797292854050128769744, 9.123148891079779403150688834967, 9.702491519458249906797937525965, 10.98626846383113899842410369322, 12.10347896070416055180365311420

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