Properties

Label 2-105-105.104-c3-0-11
Degree $2$
Conductor $105$
Sign $0.170 - 0.985i$
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  − 3.23·2-s + (3.88 + 3.45i)3-s + 2.45·4-s + (−8.12 − 7.68i)5-s + (−12.5 − 11.1i)6-s + (17.9 + 4.72i)7-s + 17.9·8-s + (3.15 + 26.8i)9-s + (26.2 + 24.8i)10-s − 0.605i·11-s + (9.52 + 8.47i)12-s − 12.8·13-s + (−57.8 − 15.2i)14-s + (−5.01 − 57.8i)15-s − 77.6·16-s + 117. i·17-s + ⋯
L(s)  = 1  − 1.14·2-s + (0.747 + 0.664i)3-s + 0.306·4-s + (−0.726 − 0.687i)5-s + (−0.854 − 0.759i)6-s + (0.966 + 0.255i)7-s + 0.792·8-s + (0.116 + 0.993i)9-s + (0.830 + 0.785i)10-s − 0.0165i·11-s + (0.229 + 0.203i)12-s − 0.273·13-s + (−1.10 − 0.291i)14-s + (−0.0863 − 0.996i)15-s − 1.21·16-s + 1.68i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.170 - 0.985i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.170 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.170 - 0.985i$
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: \((2,\ 105,\ (\ :3/2),\ 0.170 - 0.985i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.712644 + 0.599702i\)
\(L(\frac12)\) \(\approx\) \(0.712644 + 0.599702i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.88 - 3.45i)T \)
5 \( 1 + (8.12 + 7.68i)T \)
7 \( 1 + (-17.9 - 4.72i)T \)
good2 \( 1 + 3.23T + 8T^{2} \)
11 \( 1 + 0.605iT - 1.33e3T^{2} \)
13 \( 1 + 12.8T + 2.19e3T^{2} \)
17 \( 1 - 117. iT - 4.91e3T^{2} \)
19 \( 1 - 98.5iT - 6.85e3T^{2} \)
23 \( 1 - 136.T + 1.21e4T^{2} \)
29 \( 1 + 77.5iT - 2.43e4T^{2} \)
31 \( 1 - 131. iT - 2.97e4T^{2} \)
37 \( 1 - 260. iT - 5.06e4T^{2} \)
41 \( 1 - 58.0T + 6.89e4T^{2} \)
43 \( 1 + 519. iT - 7.95e4T^{2} \)
47 \( 1 + 104. iT - 1.03e5T^{2} \)
53 \( 1 - 550.T + 1.48e5T^{2} \)
59 \( 1 + 498.T + 2.05e5T^{2} \)
61 \( 1 + 172. iT - 2.26e5T^{2} \)
67 \( 1 + 622. iT - 3.00e5T^{2} \)
71 \( 1 + 151. iT - 3.57e5T^{2} \)
73 \( 1 + 242.T + 3.89e5T^{2} \)
79 \( 1 - 94.6T + 4.93e5T^{2} \)
83 \( 1 - 779. iT - 5.71e5T^{2} \)
89 \( 1 + 1.00e3T + 7.04e5T^{2} \)
97 \( 1 - 1.12e3T + 9.12e5T^{2} \)
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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

−13.63336877624294921889471936255, −12.33933332423748393255166896479, −10.98516429460492965665563197571, −10.15966764424526283092564692389, −8.854680106020198494328325270345, −8.381347336875224782756666656563, −7.57851591456059797752606098700, −5.07756511186734451138296370299, −3.95178755937964511371500404314, −1.60462095299975087616529264951, 0.77201854136246420283738917119, 2.65654426306426204374418844141, 4.57103736219579380095754899562, 7.16774556346189318633255097069, 7.45357368599713357581959639032, 8.624093332832935457136689878959, 9.494866857627333449752739792062, 10.93636102287271712191908392101, 11.64248454252332810477211983460, 13.20288709073836664540213880092

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