Properties

Label 2-105-105.104-c3-0-42
Degree $2$
Conductor $105$
Sign $-0.974 + 0.224i$
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  + 2.24·2-s + (−1.06 − 5.08i)3-s − 2.96·4-s + (−8.51 − 7.24i)5-s + (−2.39 − 11.4i)6-s + (12.2 + 13.8i)7-s − 24.6·8-s + (−24.7 + 10.8i)9-s + (−19.1 − 16.2i)10-s − 25.7i·11-s + (3.16 + 15.0i)12-s − 68.2·13-s + (27.5 + 31.1i)14-s + (−27.7 + 51.0i)15-s − 31.5·16-s − 30.6i·17-s + ⋯
L(s)  = 1  + 0.793·2-s + (−0.205 − 0.978i)3-s − 0.370·4-s + (−0.761 − 0.648i)5-s + (−0.162 − 0.776i)6-s + (0.662 + 0.748i)7-s − 1.08·8-s + (−0.915 + 0.401i)9-s + (−0.604 − 0.514i)10-s − 0.705i·11-s + (0.0760 + 0.362i)12-s − 1.45·13-s + (0.526 + 0.594i)14-s + (−0.478 + 0.878i)15-s − 0.492·16-s − 0.437i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.974 + 0.224i$
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: \((2,\ 105,\ (\ :3/2),\ -0.974 + 0.224i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.100028 - 0.880467i\)
\(L(\frac12)\) \(\approx\) \(0.100028 - 0.880467i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.06 + 5.08i)T \)
5 \( 1 + (8.51 + 7.24i)T \)
7 \( 1 + (-12.2 - 13.8i)T \)
good2 \( 1 - 2.24T + 8T^{2} \)
11 \( 1 + 25.7iT - 1.33e3T^{2} \)
13 \( 1 + 68.2T + 2.19e3T^{2} \)
17 \( 1 + 30.6iT - 4.91e3T^{2} \)
19 \( 1 + 109. iT - 6.85e3T^{2} \)
23 \( 1 - 152.T + 1.21e4T^{2} \)
29 \( 1 + 191. iT - 2.43e4T^{2} \)
31 \( 1 + 16.4iT - 2.97e4T^{2} \)
37 \( 1 + 81.9iT - 5.06e4T^{2} \)
41 \( 1 + 372.T + 6.89e4T^{2} \)
43 \( 1 + 192. iT - 7.95e4T^{2} \)
47 \( 1 - 0.366iT - 1.03e5T^{2} \)
53 \( 1 + 5.95T + 1.48e5T^{2} \)
59 \( 1 - 198.T + 2.05e5T^{2} \)
61 \( 1 + 83.5iT - 2.26e5T^{2} \)
67 \( 1 - 1.08e3iT - 3.00e5T^{2} \)
71 \( 1 + 773. iT - 3.57e5T^{2} \)
73 \( 1 - 448.T + 3.89e5T^{2} \)
79 \( 1 + 1.27e3T + 4.93e5T^{2} \)
83 \( 1 + 429. iT - 5.71e5T^{2} \)
89 \( 1 + 5.29T + 7.04e5T^{2} \)
97 \( 1 + 435.T + 9.12e5T^{2} \)
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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

−12.82405197606328126973800002231, −11.91569087508402684027820621049, −11.41195413259346570757475179046, −9.153720561649930114279106820727, −8.356757689920755744882993414956, −7.11670247895136590604202024557, −5.46129717935941805547611066404, −4.75935366893241544348865199861, −2.76696581141766306686932678939, −0.39134055827673701660836069884, 3.27567668605156166817625915330, 4.37468744296815292841652931447, 5.16137327129171854848695849492, 6.94486854714627642318027873551, 8.285373012354724123008427414310, 9.751233363834472891859132160978, 10.62518432552142824853159497609, 11.76006138756578346048807035376, 12.60424869671316495199001642711, 14.19612769828899848990719318872

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