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Properties

Label 2-105-35.19-c2-0-15
Degree $2$
Conductor $105$
Sign $0.275 + 0.961i$
Analytic cond. $2.86104$
Root an. cond. $1.69146$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.32 + 0.765i)2-s + (−0.866 − 1.5i)3-s + (−0.829 − 1.43i)4-s + (1.02 − 4.89i)5-s − 2.65i·6-s + (−6.34 − 2.94i)7-s − 8.65i·8-s + (−1.5 + 2.59i)9-s + (5.10 − 5.70i)10-s + (6.68 + 11.5i)11-s + (−1.43 + 2.48i)12-s + 15.5·13-s + (−6.15 − 8.76i)14-s + (−8.22 + 2.70i)15-s + (3.30 − 5.73i)16-s + (−10.4 − 18.0i)17-s + ⋯
L(s)  = 1  + (0.662 + 0.382i)2-s + (−0.288 − 0.5i)3-s + (−0.207 − 0.358i)4-s + (0.204 − 0.978i)5-s − 0.441i·6-s + (−0.907 − 0.421i)7-s − 1.08i·8-s + (−0.166 + 0.288i)9-s + (0.510 − 0.570i)10-s + (0.607 + 1.05i)11-s + (−0.119 + 0.207i)12-s + 1.19·13-s + (−0.439 − 0.626i)14-s + (−0.548 + 0.180i)15-s + (0.206 − 0.358i)16-s + (−0.613 − 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.275 + 0.961i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.275 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.275 + 0.961i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.275 + 0.961i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.20799 - 0.909991i\)
\(L(\frac12)\) \(\approx\) \(1.20799 - 0.909991i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 1.5i)T \)
5 \( 1 + (-1.02 + 4.89i)T \)
7 \( 1 + (6.34 + 2.94i)T \)
good2 \( 1 + (-1.32 - 0.765i)T + (2 + 3.46i)T^{2} \)
11 \( 1 + (-6.68 - 11.5i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 15.5T + 169T^{2} \)
17 \( 1 + (10.4 + 18.0i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-23.3 - 13.4i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-20.1 - 11.6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 43.9T + 841T^{2} \)
31 \( 1 + (-27.3 + 15.7i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-30.6 - 17.6i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 19.4iT - 1.68e3T^{2} \)
43 \( 1 + 18.7iT - 1.84e3T^{2} \)
47 \( 1 + (1.46 - 2.53i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (51.8 - 29.9i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-14.7 + 8.49i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-83.2 - 48.0i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-29.6 + 17.1i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 76.6T + 5.04e3T^{2} \)
73 \( 1 + (23.9 + 41.4i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (25.8 - 44.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 13.1T + 6.88e3T^{2} \)
89 \( 1 + (46.8 + 27.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 95.2T + 9.40e3T^{2} \)
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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

−13.34158198396685318885672717003, −12.67796603862259136314080537072, −11.50755180928130578306306757072, −9.826167221417760960614616235961, −9.194644885091304997122918403027, −7.35964202979267013008583127153, −6.30580325444689113782868644327, −5.23633731746413893121453272640, −3.94084871566052354283716691053, −1.08198014207306958779851975038, 3.01651475821511621211379275486, 3.81867390461840204768469057665, 5.65839117853453484979118750671, 6.60176084705510879730450997199, 8.486992489885322173269371694583, 9.506212067675471747328200169855, 11.05949714162052011134754329468, 11.37048216203264548020097053958, 12.86745394123053994507187974889, 13.58465386417705421127779691552

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