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Properties

Label 2-105-35.4-c3-0-11
Degree $2$
Conductor $105$
Sign $0.416 - 0.909i$
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  + (0.755 − 0.436i)2-s + (2.59 + 1.5i)3-s + (−3.61 + 6.26i)4-s + (10.8 − 2.81i)5-s + 2.61·6-s + (−10.7 + 15.0i)7-s + 13.2i·8-s + (4.5 + 7.79i)9-s + (6.94 − 6.84i)10-s + (10.5 − 18.2i)11-s + (−18.8 + 10.8i)12-s + 62.2i·13-s + (−1.56 + 16.0i)14-s + (32.3 + 8.92i)15-s + (−23.1 − 40.1i)16-s + (35.9 + 20.7i)17-s + ⋯
L(s)  = 1  + (0.267 − 0.154i)2-s + (0.499 + 0.288i)3-s + (−0.452 + 0.783i)4-s + (0.967 − 0.251i)5-s + 0.178·6-s + (−0.581 + 0.813i)7-s + 0.587i·8-s + (0.166 + 0.288i)9-s + (0.219 − 0.216i)10-s + (0.288 − 0.500i)11-s + (−0.452 + 0.261i)12-s + 1.32i·13-s + (−0.0298 + 0.306i)14-s + (0.556 + 0.153i)15-s + (−0.361 − 0.626i)16-s + (0.512 + 0.295i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.416 - 0.909i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.416 - 0.909i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.416 - 0.909i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.73175 + 1.11121i\)
\(L(\frac12)\) \(\approx\) \(1.73175 + 1.11121i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.59 - 1.5i)T \)
5 \( 1 + (-10.8 + 2.81i)T \)
7 \( 1 + (10.7 - 15.0i)T \)
good2 \( 1 + (-0.755 + 0.436i)T + (4 - 6.92i)T^{2} \)
11 \( 1 + (-10.5 + 18.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 62.2iT - 2.19e3T^{2} \)
17 \( 1 + (-35.9 - 20.7i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-11.2 - 19.4i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-33.8 + 19.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 59.0T + 2.43e4T^{2} \)
31 \( 1 + (-152. + 264. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (138. - 79.8i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 497.T + 6.89e4T^{2} \)
43 \( 1 + 257. iT - 7.95e4T^{2} \)
47 \( 1 + (-446. + 257. i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (36.1 + 20.8i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-136. + 235. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (371. + 642. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-661. - 381. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 42.9T + 3.57e5T^{2} \)
73 \( 1 + (104. + 60.3i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-345. - 598. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 1.23e3iT - 5.71e5T^{2} \)
89 \( 1 + (-412. - 715. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 259. iT - 9.12e5T^{2} \)
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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

−13.62550628053777075053622346169, −12.53543054385781004626179233655, −11.66820789821309956129346306130, −9.953831608573851063253039924867, −9.092431305654168173237971982465, −8.380063592171532298602582303003, −6.57462654852069286518513837644, −5.18910407485920504058515858334, −3.70742641139745606012809148437, −2.31197398501390547633393623651, 1.13582536325352695988139226549, 3.18104676965702505771381936863, 4.96761772259004185433886097588, 6.24395485794589823851344800947, 7.26033512633038255527431419729, 8.933234088354384300685255871772, 10.02721479944390744899006070594, 10.45397748550955070996763834235, 12.49295007341106056043271839359, 13.41065984437210619835657972280

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