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Properties

Label 2-105-21.20-c3-0-4
Degree $2$
Conductor $105$
Sign $0.983 + 0.179i$
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  − 5.54i·2-s + (−3.35 + 3.96i)3-s − 22.7·4-s + 5·5-s + (22.0 + 18.6i)6-s + (9.21 + 16.0i)7-s + 82.0i·8-s + (−4.49 − 26.6i)9-s − 27.7i·10-s + 18.7i·11-s + (76.4 − 90.4i)12-s − 6.49i·13-s + (89.1 − 51.1i)14-s + (−16.7 + 19.8i)15-s + 272.·16-s + 11.6·17-s + ⋯
L(s)  = 1  − 1.96i·2-s + (−0.645 + 0.763i)3-s − 2.84·4-s + 0.447·5-s + (1.49 + 1.26i)6-s + (0.497 + 0.867i)7-s + 3.62i·8-s + (−0.166 − 0.986i)9-s − 0.877i·10-s + 0.513i·11-s + (1.83 − 2.17i)12-s − 0.138i·13-s + (1.70 − 0.976i)14-s + (−0.288 + 0.341i)15-s + 4.26·16-s + 0.166·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.179i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.983 + 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.983 + 0.179i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.983 + 0.179i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.915628 - 0.0830464i\)
\(L(\frac12)\) \(\approx\) \(0.915628 - 0.0830464i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.35 - 3.96i)T \)
5 \( 1 - 5T \)
7 \( 1 + (-9.21 - 16.0i)T \)
good2 \( 1 + 5.54iT - 8T^{2} \)
11 \( 1 - 18.7iT - 1.33e3T^{2} \)
13 \( 1 + 6.49iT - 2.19e3T^{2} \)
17 \( 1 - 11.6T + 4.91e3T^{2} \)
19 \( 1 - 123. iT - 6.85e3T^{2} \)
23 \( 1 + 5.61iT - 1.21e4T^{2} \)
29 \( 1 - 174. iT - 2.43e4T^{2} \)
31 \( 1 - 213. iT - 2.97e4T^{2} \)
37 \( 1 + 176.T + 5.06e4T^{2} \)
41 \( 1 - 187.T + 6.89e4T^{2} \)
43 \( 1 - 164.T + 7.95e4T^{2} \)
47 \( 1 + 440.T + 1.03e5T^{2} \)
53 \( 1 + 144. iT - 1.48e5T^{2} \)
59 \( 1 + 49.8T + 2.05e5T^{2} \)
61 \( 1 - 308. iT - 2.26e5T^{2} \)
67 \( 1 - 97.4T + 3.00e5T^{2} \)
71 \( 1 + 491. iT - 3.57e5T^{2} \)
73 \( 1 + 787. iT - 3.89e5T^{2} \)
79 \( 1 - 824.T + 4.93e5T^{2} \)
83 \( 1 + 410.T + 5.71e5T^{2} \)
89 \( 1 + 1.61e3T + 7.04e5T^{2} \)
97 \( 1 - 747. iT - 9.12e5T^{2} \)
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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

−12.52991086097263967361437465190, −12.19832235074462968472654152975, −11.07007597502196093297382327676, −10.28903123967573134793295051114, −9.451927865719566702927093648605, −8.491299555519521583256905101527, −5.63226655672633531303705788007, −4.73002956190966127673486445274, −3.29548775757066876436123101963, −1.63027473154654082630386116967, 0.60882730081023591882265991300, 4.47881809983269800209328926577, 5.59146810787175834697641373263, 6.60266288032333089481567917777, 7.45784364349527032649914378924, 8.394161797833641561315825826390, 9.744894112189622807366671484976, 11.19508265474676650261693759196, 12.91739678397076007471128951987, 13.61732781135273935256200257605

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