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Properties

Label 2-105-7.3-c2-0-8
Degree $2$
Conductor $105$
Sign $0.987 + 0.159i$
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  + (0.288 + 0.499i)2-s + (1.5 + 0.866i)3-s + (1.83 − 3.17i)4-s + (1.93 − 1.11i)5-s + 0.999i·6-s + (−1.64 − 6.80i)7-s + 4.42·8-s + (1.5 + 2.59i)9-s + (1.11 + 0.645i)10-s + (−2.91 + 5.04i)11-s + (5.50 − 3.17i)12-s + 9.64i·13-s + (2.92 − 2.78i)14-s + 3.87·15-s + (−6.05 − 10.4i)16-s + (18.2 + 10.5i)17-s + ⋯
L(s)  = 1  + (0.144 + 0.249i)2-s + (0.5 + 0.288i)3-s + (0.458 − 0.793i)4-s + (0.387 − 0.223i)5-s + 0.166i·6-s + (−0.234 − 0.972i)7-s + 0.553·8-s + (0.166 + 0.288i)9-s + (0.111 + 0.0645i)10-s + (−0.264 + 0.458i)11-s + (0.458 − 0.264i)12-s + 0.742i·13-s + (0.209 − 0.198i)14-s + 0.258·15-s + (−0.378 − 0.655i)16-s + (1.07 + 0.621i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.987 + 0.159i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.987 + 0.159i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.86684 - 0.149449i\)
\(L(\frac12)\) \(\approx\) \(1.86684 - 0.149449i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 - 0.866i)T \)
5 \( 1 + (-1.93 + 1.11i)T \)
7 \( 1 + (1.64 + 6.80i)T \)
good2 \( 1 + (-0.288 - 0.499i)T + (-2 + 3.46i)T^{2} \)
11 \( 1 + (2.91 - 5.04i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 9.64iT - 169T^{2} \)
17 \( 1 + (-18.2 - 10.5i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (15.9 - 9.23i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-6.64 - 11.5i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 40.6T + 841T^{2} \)
31 \( 1 + (16.9 + 9.80i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (9.91 + 17.1i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 43.9iT - 1.68e3T^{2} \)
43 \( 1 + 50.8T + 1.84e3T^{2} \)
47 \( 1 + (-61.9 + 35.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (19.7 - 34.1i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-12.1 - 7.04i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-79.1 + 45.7i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-60.4 + 104. i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 66.8T + 5.04e3T^{2} \)
73 \( 1 + (-97.3 - 56.2i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-21.8 - 37.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 42.2iT - 6.88e3T^{2} \)
89 \( 1 + (-2.19 + 1.26i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 77.9iT - 9.40e3T^{2} \)
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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

−13.73479738331170925071799877847, −12.72087998308006082589409246759, −11.11401294740924191142522984832, −10.17996258158546581158895740400, −9.455981069813098186054196474252, −7.81895634441089720060928266693, −6.72407996688600913690544656468, −5.38796157208128258166493805023, −3.93949498355506265981583106269, −1.76868595713629385772921945508, 2.35883109139429358904896143023, 3.37650859881878388961148420796, 5.51097906225281222041790397266, 6.91736276194640468390707132561, 8.085139900532560264202002615990, 9.064192450496539900306974244606, 10.46609017568566388004095240385, 11.63828067266424731777958108496, 12.65339960594747393328584229528, 13.25183716324277418387361008808

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