Properties

Label 2-210-105.62-c2-0-23
Degree $2$
Conductor $210$
Sign $-0.554 + 0.832i$
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 + (2.28 − 1.94i)3-s − 2i·4-s + (−4.58 − 1.99i)5-s + (−0.347 + 4.22i)6-s + (−4.23 + 5.57i)7-s + (2 + 2i)8-s + (1.46 − 8.87i)9-s + (6.58 − 2.58i)10-s − 14.6i·11-s + (−3.88 − 4.57i)12-s + (3.48 + 3.48i)13-s + (−1.34 − 9.80i)14-s + (−14.3 + 4.32i)15-s − 4·16-s + (−20.1 − 20.1i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (0.762 − 0.646i)3-s − 0.5i·4-s + (−0.916 − 0.399i)5-s + (−0.0579 + 0.704i)6-s + (−0.604 + 0.796i)7-s + (0.250 + 0.250i)8-s + (0.163 − 0.986i)9-s + (0.658 − 0.258i)10-s − 1.32i·11-s + (−0.323 − 0.381i)12-s + (0.267 + 0.267i)13-s + (−0.0961 − 0.700i)14-s + (−0.957 + 0.288i)15-s − 0.250·16-s + (−1.18 − 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.554 + 0.832i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.554 + 0.832i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.326879 - 0.610808i\)
\(L(\frac12)\) \(\approx\) \(0.326879 - 0.610808i\)
\(L(2)\) 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 \( 1 + (-2.28 + 1.94i)T \)
5 \( 1 + (4.58 + 1.99i)T \)
7 \( 1 + (4.23 - 5.57i)T \)
good11 \( 1 + 14.6iT - 121T^{2} \)
13 \( 1 + (-3.48 - 3.48i)T + 169iT^{2} \)
17 \( 1 + (20.1 + 20.1i)T + 289iT^{2} \)
19 \( 1 + 26.4T + 361T^{2} \)
23 \( 1 + (2.68 + 2.68i)T + 529iT^{2} \)
29 \( 1 + 28.5T + 841T^{2} \)
31 \( 1 + 15.6iT - 961T^{2} \)
37 \( 1 + (-7.69 - 7.69i)T + 1.36e3iT^{2} \)
41 \( 1 - 37.9T + 1.68e3T^{2} \)
43 \( 1 + (-41.7 + 41.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (-21.0 - 21.0i)T + 2.20e3iT^{2} \)
53 \( 1 + (-47.4 - 47.4i)T + 2.80e3iT^{2} \)
59 \( 1 - 61.6iT - 3.48e3T^{2} \)
61 \( 1 + 54.1iT - 3.72e3T^{2} \)
67 \( 1 + (-68.9 - 68.9i)T + 4.48e3iT^{2} \)
71 \( 1 + 65.9iT - 5.04e3T^{2} \)
73 \( 1 + (6.51 + 6.51i)T + 5.32e3iT^{2} \)
79 \( 1 + 42.7iT - 6.24e3T^{2} \)
83 \( 1 + (-9.52 + 9.52i)T - 6.88e3iT^{2} \)
89 \( 1 + 19.3iT - 7.92e3T^{2} \)
97 \( 1 + (-84.6 + 84.6i)T - 9.40e3iT^{2} \)
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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

−11.83295382104877179406169212416, −10.93882180316268815369663022855, −9.060953171014891366929132706671, −8.951467953260646703575318575119, −7.920962967579112840004254398429, −6.84581462339016341184345122002, −5.86601350691854229551093024258, −4.04908232174414850097049131800, −2.54382855655878888112813768773, −0.39906715831325259268044644989, 2.28111055916756002234828753964, 3.81496973913636765205719745991, 4.31331270894829973311269144497, 6.72201278260505445064249645869, 7.67544234122623473158602367735, 8.588294476508762551697621085073, 9.641976610441226723746449548765, 10.59098021791425267258025289774, 11.01811772466724527231658151773, 12.61912718292521070153839027445

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