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Properties

Label 2-105-21.5-c3-0-24
Degree $2$
Conductor $105$
Sign $0.649 - 0.760i$
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  + (4.57 + 2.63i)2-s + (4.07 − 3.22i)3-s + (9.93 + 17.2i)4-s + (2.5 − 4.33i)5-s + (27.1 − 3.99i)6-s + (−17.5 + 5.84i)7-s + 62.6i·8-s + (6.18 − 26.2i)9-s + (22.8 − 13.1i)10-s + (−19.7 + 11.4i)11-s + (95.9 + 38.0i)12-s − 81.4i·13-s + (−95.7 − 19.6i)14-s + (−3.78 − 25.7i)15-s + (−85.9 + 148. i)16-s + (61.9 + 107. i)17-s + ⋯
L(s)  = 1  + (1.61 + 0.933i)2-s + (0.783 − 0.620i)3-s + (1.24 + 2.15i)4-s + (0.223 − 0.387i)5-s + (1.84 − 0.272i)6-s + (−0.948 + 0.315i)7-s + 2.77i·8-s + (0.228 − 0.973i)9-s + (0.722 − 0.417i)10-s + (−0.542 + 0.313i)11-s + (2.30 + 0.915i)12-s − 1.73i·13-s + (−1.82 − 0.375i)14-s + (−0.0651 − 0.442i)15-s + (−1.34 + 2.32i)16-s + (0.884 + 1.53i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 - 0.760i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.649 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.649 - 0.760i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.649 - 0.760i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.90421 + 1.80080i\)
\(L(\frac12)\) \(\approx\) \(3.90421 + 1.80080i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.07 + 3.22i)T \)
5 \( 1 + (-2.5 + 4.33i)T \)
7 \( 1 + (17.5 - 5.84i)T \)
good2 \( 1 + (-4.57 - 2.63i)T + (4 + 6.92i)T^{2} \)
11 \( 1 + (19.7 - 11.4i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 81.4iT - 2.19e3T^{2} \)
17 \( 1 + (-61.9 - 107. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (40.3 + 23.2i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (98.1 + 56.6i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 80.2iT - 2.43e4T^{2} \)
31 \( 1 + (15.7 - 9.06i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (49.5 - 85.7i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 142.T + 6.89e4T^{2} \)
43 \( 1 - 184.T + 7.95e4T^{2} \)
47 \( 1 + (60.6 - 105. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-200. + 115. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-3.91 - 6.77i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-158. - 91.6i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (141. + 244. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 521. iT - 3.57e5T^{2} \)
73 \( 1 + (-77.7 + 44.8i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (298. - 517. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 249.T + 5.71e5T^{2} \)
89 \( 1 + (147. - 256. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 711. iT - 9.12e5T^{2} \)
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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

−13.28460234180434755518749623255, −12.69616332921521713226342768978, −12.38754805421157125478831850340, −10.23208226789367875964987761980, −8.455697792245190331804147334745, −7.67835194810636790751637581627, −6.35793046562658186855140364729, −5.54240289876315534541235866609, −3.78474155481909577085104077675, −2.66967920088820111976081203541, 2.27147499962613947451646983951, 3.38505897563556172795583720433, 4.39419801863747809606659389877, 5.81516509682761193752946187659, 7.17725365003718564517764846989, 9.421020405169787952959307957429, 10.10384255582764229718848927757, 11.17661431088587291044869917166, 12.21429834913703589888592794804, 13.53033217605833834357990729179

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