Properties

Label 2-105-35.4-c3-0-12
Degree $2$
Conductor $105$
Sign $0.659 - 0.751i$
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  + (−4.33 + 2.5i)2-s + (2.59 + 1.5i)3-s + (8.50 − 14.7i)4-s + (−10.5 + 3.76i)5-s − 15.0·6-s + (15.5 − 10i)7-s + 45.0i·8-s + (4.5 + 7.79i)9-s + (36.1 − 42.6i)10-s + (22.5 − 38.9i)11-s + (44.1 − 25.5i)12-s − 37i·13-s + (−42.5 + 82.2i)14-s + (−33 − 6i)15-s + (−44.5 − 77.0i)16-s + (50.2 + 29i)17-s + ⋯
L(s)  = 1  + (−1.53 + 0.883i)2-s + (0.499 + 0.288i)3-s + (1.06 − 1.84i)4-s + (−0.941 + 0.337i)5-s − 1.02·6-s + (0.841 − 0.539i)7-s + 1.98i·8-s + (0.166 + 0.288i)9-s + (1.14 − 1.34i)10-s + (0.616 − 1.06i)11-s + (1.06 − 0.613i)12-s − 0.789i·13-s + (−0.811 + 1.57i)14-s + (−0.568 − 0.103i)15-s + (−0.695 − 1.20i)16-s + (0.716 + 0.413i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.659 - 0.751i$
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: \((2,\ 105,\ (\ :3/2),\ 0.659 - 0.751i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.780423 + 0.353557i\)
\(L(\frac12)\) \(\approx\) \(0.780423 + 0.353557i\)
\(L(\frac{5}{2})\) 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 \( 1 + (10.5 - 3.76i)T \)
7 \( 1 + (-15.5 + 10i)T \)
good2 \( 1 + (4.33 - 2.5i)T + (4 - 6.92i)T^{2} \)
11 \( 1 + (-22.5 + 38.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 37iT - 2.19e3T^{2} \)
17 \( 1 + (-50.2 - 29i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-67.5 - 116. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (54.5 - 31.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 184T + 2.43e4T^{2} \)
31 \( 1 + (85 - 147. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-61.4 + 35.5i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 247T + 6.89e4T^{2} \)
43 \( 1 + 242iT - 7.95e4T^{2} \)
47 \( 1 + (-491. + 283.5i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (38.9 + 22.5i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (131 + 226. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-756. - 437i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 1.13e3T + 3.57e5T^{2} \)
73 \( 1 + (-93.5 - 54i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (31 + 53.6i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 454iT - 5.71e5T^{2} \)
89 \( 1 + (603 + 1.04e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.36e3iT - 9.12e5T^{2} \)
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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

−14.14749495064649157592627549208, −11.97579898195771153513794443896, −10.78500750863729656835927031527, −10.14237003094837043539063497045, −8.657584372759447094067262289412, −8.035896189564581621594601153064, −7.30090253369165016912246666507, −5.73711241418096496789958973955, −3.66469536531814784122355287298, −1.02707373728464075859609208493, 1.13750445235385940111755474844, 2.59974059057939210308644225453, 4.41481219152517174749296249440, 7.17313733947932755363815623258, 7.896503545722239773498890309907, 8.955197920804585691852598522289, 9.573325932325800565805575918598, 11.15325109662854079803347684974, 11.86061438562012363067360899602, 12.47062224613502589408747680155

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