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Properties

Label 2-105-5.4-c3-0-3
Degree $2$
Conductor $105$
Sign $-0.588 + 0.808i$
Analytic cond. $6.19520$
Root an. cond. $2.48901$
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  + 4.10i·2-s + 3i·3-s − 8.86·4-s + (−6.58 + 9.03i)5-s − 12.3·6-s − 7i·7-s − 3.54i·8-s − 9·9-s + (−37.1 − 27.0i)10-s − 8.17·11-s − 26.5i·12-s − 19.2i·13-s + 28.7·14-s + (−27.1 − 19.7i)15-s − 56.3·16-s + 18.8i·17-s + ⋯
L(s)  = 1  + 1.45i·2-s + 0.577i·3-s − 1.10·4-s + (−0.588 + 0.808i)5-s − 0.838·6-s − 0.377i·7-s − 0.156i·8-s − 0.333·9-s + (−1.17 − 0.854i)10-s − 0.224·11-s − 0.639i·12-s − 0.410i·13-s + 0.548·14-s + (−0.466 − 0.339i)15-s − 0.880·16-s + 0.269i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 + 0.808i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.588 + 0.808i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ -0.588 + 0.808i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.453370 - 0.890990i\)
\(L(\frac12)\) \(\approx\) \(0.453370 - 0.890990i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3iT \)
5 \( 1 + (6.58 - 9.03i)T \)
7 \( 1 + 7iT \)
good2 \( 1 - 4.10iT - 8T^{2} \)
11 \( 1 + 8.17T + 1.33e3T^{2} \)
13 \( 1 + 19.2iT - 2.19e3T^{2} \)
17 \( 1 - 18.8iT - 4.91e3T^{2} \)
19 \( 1 - 76.5T + 6.85e3T^{2} \)
23 \( 1 - 142. iT - 1.21e4T^{2} \)
29 \( 1 - 96.1T + 2.43e4T^{2} \)
31 \( 1 + 270.T + 2.97e4T^{2} \)
37 \( 1 - 335. iT - 5.06e4T^{2} \)
41 \( 1 + 122.T + 6.89e4T^{2} \)
43 \( 1 - 492. iT - 7.95e4T^{2} \)
47 \( 1 - 96.9iT - 1.03e5T^{2} \)
53 \( 1 - 388. iT - 1.48e5T^{2} \)
59 \( 1 + 112.T + 2.05e5T^{2} \)
61 \( 1 - 347.T + 2.26e5T^{2} \)
67 \( 1 + 101. iT - 3.00e5T^{2} \)
71 \( 1 - 304.T + 3.57e5T^{2} \)
73 \( 1 + 753. iT - 3.89e5T^{2} \)
79 \( 1 - 1.16e3T + 4.93e5T^{2} \)
83 \( 1 + 889. iT - 5.71e5T^{2} \)
89 \( 1 - 938.T + 7.04e5T^{2} \)
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

−14.36488020088111174621361177844, −13.42286148699921995267441326430, −11.69420755533022009401591765164, −10.71805556058653586469142037243, −9.490945871799118168457222995159, −8.064652929417968694630220730156, −7.35917209987733864592117535795, −6.15250014484073278804618591728, −4.89684779436448127490688734504, −3.36213173903108390410838597236, 0.55671692056768989432787232444, 2.13597506937115467038971821028, 3.72134234524095118230410321873, 5.19410589850972272120247348723, 7.09133574342786948010398918269, 8.524926695296684038078188186242, 9.415557216374253421768864985570, 10.78615405625444198851283405598, 11.77075002084772584406948682843, 12.38025684974231555064837284956

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