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Properties

Label 2-105-105.104-c3-0-40
Degree $2$
Conductor $105$
Sign $0.723 + 0.689i$
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  + 5.11·2-s + (1.51 − 4.96i)3-s + 18.1·4-s + (−10.9 + 2.25i)5-s + (7.75 − 25.4i)6-s + (11.4 − 14.5i)7-s + 51.7·8-s + (−22.3 − 15.0i)9-s + (−55.9 + 11.5i)10-s + 55.4i·11-s + (27.5 − 90.0i)12-s + 20.9·13-s + (58.6 − 74.3i)14-s + (−5.42 + 57.8i)15-s + 119.·16-s + 96.8i·17-s + ⋯
L(s)  = 1  + 1.80·2-s + (0.292 − 0.956i)3-s + 2.26·4-s + (−0.979 + 0.201i)5-s + (0.527 − 1.72i)6-s + (0.619 − 0.785i)7-s + 2.28·8-s + (−0.829 − 0.558i)9-s + (−1.77 + 0.364i)10-s + 1.51i·11-s + (0.661 − 2.16i)12-s + 0.447·13-s + (1.11 − 1.41i)14-s + (−0.0933 + 0.995i)15-s + 1.86·16-s + 1.38i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 + 0.689i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.723 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.723 + 0.689i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.723 + 0.689i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.85782 - 1.54388i\)
\(L(\frac12)\) \(\approx\) \(3.85782 - 1.54388i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.51 + 4.96i)T \)
5 \( 1 + (10.9 - 2.25i)T \)
7 \( 1 + (-11.4 + 14.5i)T \)
good2 \( 1 - 5.11T + 8T^{2} \)
11 \( 1 - 55.4iT - 1.33e3T^{2} \)
13 \( 1 - 20.9T + 2.19e3T^{2} \)
17 \( 1 - 96.8iT - 4.91e3T^{2} \)
19 \( 1 + 33.1iT - 6.85e3T^{2} \)
23 \( 1 + 111.T + 1.21e4T^{2} \)
29 \( 1 + 156. iT - 2.43e4T^{2} \)
31 \( 1 - 80.4iT - 2.97e4T^{2} \)
37 \( 1 - 180. iT - 5.06e4T^{2} \)
41 \( 1 - 36.5T + 6.89e4T^{2} \)
43 \( 1 + 52.5iT - 7.95e4T^{2} \)
47 \( 1 + 259. iT - 1.03e5T^{2} \)
53 \( 1 - 191.T + 1.48e5T^{2} \)
59 \( 1 + 705.T + 2.05e5T^{2} \)
61 \( 1 + 427. iT - 2.26e5T^{2} \)
67 \( 1 - 306. iT - 3.00e5T^{2} \)
71 \( 1 + 513. iT - 3.57e5T^{2} \)
73 \( 1 - 360.T + 3.89e5T^{2} \)
79 \( 1 - 85.9T + 4.93e5T^{2} \)
83 \( 1 + 886. iT - 5.71e5T^{2} \)
89 \( 1 - 1.41e3T + 7.04e5T^{2} \)
97 \( 1 + 1.05e3T + 9.12e5T^{2} \)
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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

−13.17278340552546137219583661903, −12.31196173413618097476999916416, −11.65527240502144755547529699782, −10.55652292838988808270700014507, −8.093876770510132849388663761267, −7.26788801605883563962949489141, −6.32402080198321491939071275218, −4.60167228455836344990584445739, −3.65095594435063346759079986656, −1.91738739532502118780986923318, 2.87619937840653828452262971637, 3.88453353614480999492676831189, 5.01556319409932981095049214203, 5.92381355545838575276169956187, 7.80518544758066760367780186332, 8.918720598066300922292844187377, 10.94783555577813583521417495541, 11.44251333124204226068064736513, 12.31184794024894174380315990300, 13.74920438710522171491484118849

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