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Properties

Label 2-105-35.2-c2-0-9
Degree $2$
Conductor $105$
Sign $0.258 + 0.966i$
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  + (−0.445 + 0.119i)2-s + (−1.67 − 0.448i)3-s + (−3.28 + 1.89i)4-s + (4.59 − 1.97i)5-s + 0.798·6-s + (−0.171 − 6.99i)7-s + (2.53 − 2.53i)8-s + (2.59 + 1.50i)9-s + (−1.81 + 1.42i)10-s + (−7.70 − 13.3i)11-s + (6.33 − 1.69i)12-s + (17.1 − 17.1i)13-s + (0.911 + 3.09i)14-s + (−8.57 + 1.23i)15-s + (6.74 − 11.6i)16-s + (−5.59 + 20.8i)17-s + ⋯
L(s)  = 1  + (−0.222 + 0.0596i)2-s + (−0.557 − 0.149i)3-s + (−0.820 + 0.473i)4-s + (0.919 − 0.394i)5-s + 0.133·6-s + (−0.0245 − 0.999i)7-s + (0.317 − 0.317i)8-s + (0.288 + 0.166i)9-s + (−0.181 + 0.142i)10-s + (−0.700 − 1.21i)11-s + (0.528 − 0.141i)12-s + (1.31 − 1.31i)13-s + (0.0650 + 0.221i)14-s + (−0.571 + 0.0823i)15-s + (0.421 − 0.730i)16-s + (−0.329 + 1.22i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.966i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.258 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.258 + 0.966i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.258 + 0.966i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.690587 - 0.530296i\)
\(L(\frac12)\) \(\approx\) \(0.690587 - 0.530296i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.67 + 0.448i)T \)
5 \( 1 + (-4.59 + 1.97i)T \)
7 \( 1 + (0.171 + 6.99i)T \)
good2 \( 1 + (0.445 - 0.119i)T + (3.46 - 2i)T^{2} \)
11 \( 1 + (7.70 + 13.3i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-17.1 + 17.1i)T - 169iT^{2} \)
17 \( 1 + (5.59 - 20.8i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (15.6 + 9.01i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-2.98 - 11.1i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + 1.87iT - 841T^{2} \)
31 \( 1 + (-9.52 - 16.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (4.95 - 1.32i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 51.4T + 1.68e3T^{2} \)
43 \( 1 + (-18.4 + 18.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (-22.4 + 6.01i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-75.2 - 20.1i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (40.1 - 23.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-12.3 + 21.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-29.9 + 111. i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 63.9T + 5.04e3T^{2} \)
73 \( 1 + (-13.3 - 3.56i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-45.3 - 26.1i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (34.3 - 34.3i)T - 6.88e3iT^{2} \)
89 \( 1 + (-44.5 - 25.7i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-39.0 - 39.0i)T + 9.40e3iT^{2} \)
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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

−13.37519405889593119022513047896, −12.67411486560104114301918179534, −10.77709160506198804602463660550, −10.37123037152715418990400139333, −8.771372547851362957884049297503, −8.033400590797985317076121496343, −6.32407343478895803608733192632, −5.22092668106250627096254968859, −3.65279742462383860859464010055, −0.805288444918178423838659194429, 2.01974009274870469289009725214, 4.55366778364991520024698907387, 5.65165017314797436258241683480, 6.71518510514714561773045608814, 8.686891347512388182689427973348, 9.521213368103004550472348721514, 10.38106897048977958179552375432, 11.50122686121874639722826758489, 12.81190995999610631614614590834, 13.70676949874457052738246623813

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