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Properties

Label 2-105-35.12-c1-0-7
Degree $2$
Conductor $105$
Sign $0.761 + 0.647i$
Analytic cond. $0.838429$
Root an. cond. $0.915657$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.72 − 0.461i)2-s + (0.258 − 0.965i)3-s + (1.01 − 0.587i)4-s + (−1.95 − 1.09i)5-s − 1.78i·6-s + (1.68 + 2.04i)7-s + (−1.03 + 1.03i)8-s + (−0.866 − 0.499i)9-s + (−3.86 − 0.983i)10-s + (1.46 + 2.54i)11-s + (−0.304 − 1.13i)12-s + (−0.187 − 0.187i)13-s + (3.83 + 2.74i)14-s + (−1.56 + 1.60i)15-s + (−2.48 + 4.30i)16-s + (−3.24 − 0.868i)17-s + ⋯
L(s)  = 1  + (1.21 − 0.326i)2-s + (0.149 − 0.557i)3-s + (0.508 − 0.293i)4-s + (−0.872 − 0.489i)5-s − 0.727i·6-s + (0.635 + 0.772i)7-s + (−0.367 + 0.367i)8-s + (−0.288 − 0.166i)9-s + (−1.22 − 0.310i)10-s + (0.442 + 0.766i)11-s + (−0.0877 − 0.327i)12-s + (−0.0521 − 0.0521i)13-s + (1.02 + 0.732i)14-s + (−0.403 + 0.413i)15-s + (−0.621 + 1.07i)16-s + (−0.786 − 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 + 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.761 + 0.647i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.761 + 0.647i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52534 - 0.561051i\)
\(L(\frac12)\) \(\approx\) \(1.52534 - 0.561051i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (1.95 + 1.09i)T \)
7 \( 1 + (-1.68 - 2.04i)T \)
good2 \( 1 + (-1.72 + 0.461i)T + (1.73 - i)T^{2} \)
11 \( 1 + (-1.46 - 2.54i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.187 + 0.187i)T + 13iT^{2} \)
17 \( 1 + (3.24 + 0.868i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.81 + 3.14i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.43 + 9.07i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 0.815iT - 29T^{2} \)
31 \( 1 + (3.76 - 2.17i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.31 + 1.69i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 2.45iT - 41T^{2} \)
43 \( 1 + (3.59 - 3.59i)T - 43iT^{2} \)
47 \( 1 + (2.47 + 9.24i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-4.87 - 1.30i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-5.41 - 9.37i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.07 - 4.66i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.10 + 15.3i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.68T + 71T^{2} \)
73 \( 1 + (-0.508 + 1.89i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (4.44 + 2.56i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.09 - 6.09i)T + 83iT^{2} \)
89 \( 1 + (4.87 - 8.43i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.93 - 5.93i)T - 97iT^{2} \)
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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

−13.45383133340990626642135792801, −12.52973313500285284532923721513, −11.96049468546450864994888854727, −11.15769310695633550460475890254, −9.051307387651882413384464313377, −8.190703544298217003423159377793, −6.71243749536483423696921519790, −5.14201248000195754886039253861, −4.20567262394027994846187201549, −2.43554429094350330114349715867, 3.52170469468353507781913000235, 4.19623557115056010042015351607, 5.59070673171755805795613696492, 7.00380288704052513510646648934, 8.179911074345465462534310319397, 9.706346229858171782248084553620, 11.13262083921718171767333408197, 11.70419630932685241206569519710, 13.18804069863952416056829085612, 14.12216636710853345902714394671

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