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Properties

Label 2-105-21.20-c3-0-2
Degree $2$
Conductor $105$
Sign $-0.314 + 0.949i$
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  + 4.47i·2-s + (−0.787 + 5.13i)3-s − 12.0·4-s + 5·5-s + (−22.9 − 3.52i)6-s + (−18.2 − 3.09i)7-s − 18.1i·8-s + (−25.7 − 8.08i)9-s + 22.3i·10-s − 29.0i·11-s + (9.48 − 61.8i)12-s + 71.4i·13-s + (13.8 − 81.7i)14-s + (−3.93 + 25.6i)15-s − 15.2·16-s + 55.3·17-s + ⋯
L(s)  = 1  + 1.58i·2-s + (−0.151 + 0.988i)3-s − 1.50·4-s + 0.447·5-s + (−1.56 − 0.239i)6-s + (−0.985 − 0.167i)7-s − 0.801i·8-s + (−0.954 − 0.299i)9-s + 0.707i·10-s − 0.794i·11-s + (0.228 − 1.48i)12-s + 1.52i·13-s + (0.264 − 1.56i)14-s + (−0.0677 + 0.442i)15-s − 0.237·16-s + 0.789·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.314 + 0.949i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.314 + 0.949i$
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: \((2,\ 105,\ (\ :3/2),\ -0.314 + 0.949i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.539616 - 0.747521i\)
\(L(\frac12)\) \(\approx\) \(0.539616 - 0.747521i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.787 - 5.13i)T \)
5 \( 1 - 5T \)
7 \( 1 + (18.2 + 3.09i)T \)
good2 \( 1 - 4.47iT - 8T^{2} \)
11 \( 1 + 29.0iT - 1.33e3T^{2} \)
13 \( 1 - 71.4iT - 2.19e3T^{2} \)
17 \( 1 - 55.3T + 4.91e3T^{2} \)
19 \( 1 + 3.58iT - 6.85e3T^{2} \)
23 \( 1 - 80.1iT - 1.21e4T^{2} \)
29 \( 1 - 177. iT - 2.43e4T^{2} \)
31 \( 1 - 66.1iT - 2.97e4T^{2} \)
37 \( 1 + 353.T + 5.06e4T^{2} \)
41 \( 1 + 329.T + 6.89e4T^{2} \)
43 \( 1 + 70.6T + 7.95e4T^{2} \)
47 \( 1 - 199.T + 1.03e5T^{2} \)
53 \( 1 - 504. iT - 1.48e5T^{2} \)
59 \( 1 - 392.T + 2.05e5T^{2} \)
61 \( 1 - 724. iT - 2.26e5T^{2} \)
67 \( 1 - 411.T + 3.00e5T^{2} \)
71 \( 1 + 1.00e3iT - 3.57e5T^{2} \)
73 \( 1 + 341. iT - 3.89e5T^{2} \)
79 \( 1 - 184.T + 4.93e5T^{2} \)
83 \( 1 - 1.33e3T + 5.71e5T^{2} \)
89 \( 1 + 1.51e3T + 7.04e5T^{2} \)
97 \( 1 - 689. iT - 9.12e5T^{2} \)
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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

−14.12820657079179113564463998474, −13.66372178632946022292352775155, −11.93658866443277438389164883710, −10.50934969450352914525055598187, −9.339373162464900929210071456274, −8.722890456347404742295983604690, −7.04001349469451659179462089506, −6.09548927290276327480559908064, −5.10618252313742296589313161692, −3.57538109120359671005392813931, 0.51274590362511274684400427141, 2.17079037700110422351761309623, 3.31530862260674595351948052987, 5.41983073035254988104108613795, 6.82010735729189219606948553703, 8.353711177453662194615491256690, 9.791348099839290463820262425787, 10.38760766660027589095642653504, 11.74746538241537770323099566083, 12.63772956251172055441510046940

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