Properties

Label 2-105-7.4-c3-0-2
Degree $2$
Conductor $105$
Sign $-0.872 + 0.487i$
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  + (2.40 + 4.16i)2-s + (−1.5 + 2.59i)3-s + (−7.55 + 13.0i)4-s + (−2.5 − 4.33i)5-s − 14.4·6-s + (−18.4 + 0.916i)7-s − 34.1·8-s + (−4.5 − 7.79i)9-s + (12.0 − 20.8i)10-s + (−26.6 + 46.1i)11-s + (−22.6 − 39.2i)12-s + 60.9·13-s + (−48.2 − 74.7i)14-s + 15.0·15-s + (−21.6 − 37.4i)16-s + (−15.5 + 26.9i)17-s + ⋯
L(s)  = 1  + (0.849 + 1.47i)2-s + (−0.288 + 0.499i)3-s + (−0.943 + 1.63i)4-s + (−0.223 − 0.387i)5-s − 0.981·6-s + (−0.998 + 0.0494i)7-s − 1.50·8-s + (−0.166 − 0.288i)9-s + (0.379 − 0.658i)10-s + (−0.729 + 1.26i)11-s + (−0.544 − 0.943i)12-s + 1.30·13-s + (−0.921 − 1.42i)14-s + 0.258·15-s + (−0.337 − 0.585i)16-s + (−0.221 + 0.384i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.487i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.872 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.872 + 0.487i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ -0.872 + 0.487i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.377669 - 1.44965i\)
\(L(\frac12)\) \(\approx\) \(0.377669 - 1.44965i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 2.59i)T \)
5 \( 1 + (2.5 + 4.33i)T \)
7 \( 1 + (18.4 - 0.916i)T \)
good2 \( 1 + (-2.40 - 4.16i)T + (-4 + 6.92i)T^{2} \)
11 \( 1 + (26.6 - 46.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 60.9T + 2.19e3T^{2} \)
17 \( 1 + (15.5 - 26.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-8.95 - 15.5i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-17.4 - 30.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 141.T + 2.43e4T^{2} \)
31 \( 1 + (58.8 - 101. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (87.6 + 151. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 411.T + 6.89e4T^{2} \)
43 \( 1 + 498.T + 7.95e4T^{2} \)
47 \( 1 + (-145. - 251. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (291. - 504. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-328. + 569. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-208. - 361. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-283. + 491. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 887.T + 3.57e5T^{2} \)
73 \( 1 + (-546. + 946. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (67.7 + 117. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 464.T + 5.71e5T^{2} \)
89 \( 1 + (15.9 + 27.6i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 254.T + 9.12e5T^{2} \)
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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

−14.00408784802067505109338326227, −12.96676785809830311999938226268, −12.39388816685275620276085056648, −10.66466468041850964239655199538, −9.334366802545653214089285992936, −8.133086704706312509812816400832, −6.88639463329584295293308659203, −5.89233683105249080949113231565, −4.75716380527812804619769674177, −3.62019457042287640956862342666, 0.67125472143038036290580231823, 2.72021712981761545904342377002, 3.70739930034245205155414830905, 5.47938895223218035768753701164, 6.59990432187918655558864940167, 8.453092919080817799711737403654, 9.979515328718235859972409937125, 10.97206925789305312520573224802, 11.53680685651777820126431266142, 12.77765631489496752758317992951

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