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Properties

Label 2-105-35.3-c1-0-7
Degree $2$
Conductor $105$
Sign $-0.615 + 0.787i$
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  + (−0.394 − 0.105i)2-s + (−0.258 − 0.965i)3-s + (−1.58 − 0.916i)4-s + (−2.18 − 0.456i)5-s + 0.408i·6-s + (−0.605 − 2.57i)7-s + (1.10 + 1.10i)8-s + (−0.866 + 0.499i)9-s + (0.815 + 0.411i)10-s + (−0.463 + 0.803i)11-s + (−0.474 + 1.77i)12-s + (4.08 − 4.08i)13-s + (−0.0332 + 1.08i)14-s + (0.125 + 2.23i)15-s + (1.51 + 2.62i)16-s + (−0.719 + 0.192i)17-s + ⋯
L(s)  = 1  + (−0.278 − 0.0747i)2-s + (−0.149 − 0.557i)3-s + (−0.793 − 0.458i)4-s + (−0.978 − 0.204i)5-s + 0.166i·6-s + (−0.228 − 0.973i)7-s + (0.391 + 0.391i)8-s + (−0.288 + 0.166i)9-s + (0.257 + 0.130i)10-s + (−0.139 + 0.242i)11-s + (−0.136 + 0.511i)12-s + (1.13 − 1.13i)13-s + (−0.00889 + 0.288i)14-s + (0.0324 + 0.576i)15-s + (0.378 + 0.655i)16-s + (−0.174 + 0.0467i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 + 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.615 + 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.615 + 0.787i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ -0.615 + 0.787i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.232449 - 0.476607i\)
\(L(\frac12)\) \(\approx\) \(0.232449 - 0.476607i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.18 + 0.456i)T \)
7 \( 1 + (0.605 + 2.57i)T \)
good2 \( 1 + (0.394 + 0.105i)T + (1.73 + i)T^{2} \)
11 \( 1 + (0.463 - 0.803i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.08 + 4.08i)T - 13iT^{2} \)
17 \( 1 + (0.719 - 0.192i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.21 + 2.11i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.34 + 5.00i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 8.08iT - 29T^{2} \)
31 \( 1 + (1.05 + 0.607i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.76 - 0.472i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 6.97iT - 41T^{2} \)
43 \( 1 + (0.781 + 0.781i)T + 43iT^{2} \)
47 \( 1 + (-2.70 + 10.0i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.42 - 1.72i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.91 + 10.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.72 - 2.15i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.80 - 10.4i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 9.89T + 71T^{2} \)
73 \( 1 + (-1.07 - 4.02i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-7.02 + 4.05i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.91 - 5.91i)T - 83iT^{2} \)
89 \( 1 + (-7.78 - 13.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.89 - 4.89i)T + 97iT^{2} \)
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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

−13.21189813410692581451734981472, −12.64472911150245213040715616328, −11.02001584589905962015506216063, −10.45534880656640427504425821708, −8.833185265165351180452971500887, −8.009313101691063178869474435593, −6.78630259431665824479856703114, −5.11890536765136991989483194714, −3.74865933430354006370140058281, −0.73023617515681442252256110184, 3.44233775304889305593324717385, 4.52693550130699163652192726595, 6.17343951978471211264013787585, 7.85345616637111682704192847553, 8.778480047348444281923558309949, 9.596944778483121876475350019486, 11.13952701899886353300013407608, 11.88995870497709348262480530675, 13.04966768108686961350159743512, 14.18868957529387693655448299606

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