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Properties

Label 2-105-105.59-c1-0-10
Degree $2$
Conductor $105$
Sign $0.517 + 0.855i$
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  + (0.322 + 0.558i)2-s + (−1.15 − 1.28i)3-s + (0.792 − 1.37i)4-s + (−0.550 − 2.16i)5-s + (0.346 − 1.06i)6-s + (−2.64 − 0.105i)7-s + 2.31·8-s + (−0.319 + 2.98i)9-s + (1.03 − 1.00i)10-s + (3.51 + 2.02i)11-s + (−2.68 + 0.567i)12-s + 4.21·13-s + (−0.793 − 1.51i)14-s + (−2.15 + 3.21i)15-s + (−0.839 − 1.45i)16-s + (1.88 + 1.08i)17-s + ⋯
L(s)  = 1  + (0.227 + 0.394i)2-s + (−0.668 − 0.743i)3-s + (0.396 − 0.685i)4-s + (−0.246 − 0.969i)5-s + (0.141 − 0.433i)6-s + (−0.999 − 0.0397i)7-s + 0.817·8-s + (−0.106 + 0.994i)9-s + (0.326 − 0.318i)10-s + (1.05 + 0.611i)11-s + (−0.774 + 0.163i)12-s + 1.16·13-s + (−0.212 − 0.403i)14-s + (−0.556 + 0.830i)15-s + (−0.209 − 0.363i)16-s + (0.457 + 0.263i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.517 + 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.517 + 0.855i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.517 + 0.855i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.850646 - 0.479642i\)
\(L(\frac12)\) \(\approx\) \(0.850646 - 0.479642i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.15 + 1.28i)T \)
5 \( 1 + (0.550 + 2.16i)T \)
7 \( 1 + (2.64 + 0.105i)T \)
good2 \( 1 + (-0.322 - 0.558i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (-3.51 - 2.02i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.21T + 13T^{2} \)
17 \( 1 + (-1.88 - 1.08i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.87 - 2.23i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.322 + 0.558i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.16iT - 29T^{2} \)
31 \( 1 + (0.339 + 0.195i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.69 + 2.13i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.27T + 41T^{2} \)
43 \( 1 - 6.54iT - 43T^{2} \)
47 \( 1 + (6.75 - 3.90i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.60 - 6.23i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.66 - 9.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.05 + 3.49i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.56 + 4.36i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.13iT - 71T^{2} \)
73 \( 1 + (2.61 - 4.53i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.87 + 3.24i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.27iT - 83T^{2} \)
89 \( 1 + (0.447 + 0.774i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.89T + 97T^{2} \)
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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

−13.40900658602365553342313966600, −12.65270840319415574697817888160, −11.70196332433311297692512836622, −10.54288030495951722297130835233, −9.282811194284658584105336219906, −7.81124731789452664396494128916, −6.47113962505126988678222195331, −5.90943191946909424416573825390, −4.33402355750316525292965349509, −1.39281036532228703720151054819, 3.22454018178035205889359300612, 3.95989342481164444408658015231, 6.15757224532687528355769937967, 6.87039414161545320405319029065, 8.654775394482522614803922911242, 10.00426470295355964489685876850, 11.06317439046278664099960305540, 11.55989913551281207024739939031, 12.67773869471600451376523390110, 13.85028814861969750882002884277

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