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Properties

Label 2-105-105.62-c2-0-14
Degree $2$
Conductor $105$
Sign $0.790 + 0.612i$
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  + (−2.28 + 2.28i)2-s + (−1.07 + 2.80i)3-s − 6.40i·4-s + (1.80 − 4.66i)5-s + (−3.93 − 8.83i)6-s + (−6.22 − 3.20i)7-s + (5.48 + 5.48i)8-s + (−6.69 − 6.01i)9-s + (6.50 + 14.7i)10-s − 11.1i·11-s + (17.9 + 6.88i)12-s + (5.82 + 5.82i)13-s + (21.5 − 6.86i)14-s + (11.1 + 10.0i)15-s + 0.592·16-s + (6.84 + 6.84i)17-s + ⋯
L(s)  = 1  + (−1.14 + 1.14i)2-s + (−0.358 + 0.933i)3-s − 1.60i·4-s + (0.361 − 0.932i)5-s + (−0.656 − 1.47i)6-s + (−0.888 − 0.458i)7-s + (0.685 + 0.685i)8-s + (−0.743 − 0.668i)9-s + (0.650 + 1.47i)10-s − 1.01i·11-s + (1.49 + 0.573i)12-s + (0.448 + 0.448i)13-s + (1.53 − 0.490i)14-s + (0.740 + 0.671i)15-s + 0.0370·16-s + (0.402 + 0.402i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.790 + 0.612i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.790 + 0.612i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.316757 - 0.108374i\)
\(L(\frac12)\) \(\approx\) \(0.316757 - 0.108374i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.07 - 2.80i)T \)
5 \( 1 + (-1.80 + 4.66i)T \)
7 \( 1 + (6.22 + 3.20i)T \)
good2 \( 1 + (2.28 - 2.28i)T - 4iT^{2} \)
11 \( 1 + 11.1iT - 121T^{2} \)
13 \( 1 + (-5.82 - 5.82i)T + 169iT^{2} \)
17 \( 1 + (-6.84 - 6.84i)T + 289iT^{2} \)
19 \( 1 + 25.0T + 361T^{2} \)
23 \( 1 + (23.3 + 23.3i)T + 529iT^{2} \)
29 \( 1 - 10.6T + 841T^{2} \)
31 \( 1 + 26.9iT - 961T^{2} \)
37 \( 1 + (20.8 + 20.8i)T + 1.36e3iT^{2} \)
41 \( 1 - 32.9T + 1.68e3T^{2} \)
43 \( 1 + (-1.25 + 1.25i)T - 1.84e3iT^{2} \)
47 \( 1 + (59.1 + 59.1i)T + 2.20e3iT^{2} \)
53 \( 1 + (-26.0 - 26.0i)T + 2.80e3iT^{2} \)
59 \( 1 - 70.2iT - 3.48e3T^{2} \)
61 \( 1 + 14.1iT - 3.72e3T^{2} \)
67 \( 1 + (6.14 + 6.14i)T + 4.48e3iT^{2} \)
71 \( 1 + 39.0iT - 5.04e3T^{2} \)
73 \( 1 + (51.1 + 51.1i)T + 5.32e3iT^{2} \)
79 \( 1 - 16.8iT - 6.24e3T^{2} \)
83 \( 1 + (31.3 - 31.3i)T - 6.88e3iT^{2} \)
89 \( 1 - 70.8iT - 7.92e3T^{2} \)
97 \( 1 + (114. - 114. i)T - 9.40e3iT^{2} \)
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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

−13.68735122459342608897275223050, −12.34501149345211348937499658347, −10.70175417218389912798897786794, −9.919281738064346247157895869514, −8.946134196615351254338342898426, −8.293067537942761375803907766796, −6.41792335236287531883983700983, −5.80765555511973259586416325511, −4.05965943790312508621181356171, −0.36324139024709615945713561161, 1.89711609327720206838822880085, 3.07777932648089067216142118021, 5.96180207190513305514522380492, 7.10681961836098273881963238698, 8.313212414828118838160900083060, 9.659146273287731680338360465122, 10.39982095847710024432405318213, 11.43702603348641561641457297095, 12.36212661154777173417101184993, 13.10487080764373714713260924464

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