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Properties

Label 2-105-35.17-c1-0-1
Degree $2$
Conductor $105$
Sign $-0.263 - 0.964i$
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  + (−0.461 + 1.72i)2-s + (0.965 − 0.258i)3-s + (−1.01 − 0.587i)4-s + (−1.92 + 1.14i)5-s + 1.78i·6-s + (2.04 + 1.68i)7-s + (−1.03 + 1.03i)8-s + (0.866 − 0.499i)9-s + (−1.07 − 3.83i)10-s + (1.46 − 2.54i)11-s + (−1.13 − 0.304i)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 + (−0.868 − 3.24i)17-s + ⋯
L(s)  = 1  + (−0.326 + 1.21i)2-s + (0.557 − 0.149i)3-s + (−0.508 − 0.293i)4-s + (−0.859 + 0.510i)5-s + 0.727i·6-s + (0.772 + 0.635i)7-s + (−0.367 + 0.367i)8-s + (0.288 − 0.166i)9-s + (−0.341 − 1.21i)10-s + (0.442 − 0.766i)11-s + (−0.327 − 0.0877i)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 + (−0.210 − 0.786i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.263 - 0.964i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ -0.263 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.598465 + 0.783553i\)
\(L(\frac12)\) \(\approx\) \(0.598465 + 0.783553i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (1.92 - 1.14i)T \)
7 \( 1 + (-2.04 - 1.68i)T \)
good2 \( 1 + (0.461 - 1.72i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (-1.46 + 2.54i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.187 - 0.187i)T + 13iT^{2} \)
17 \( 1 + (0.868 + 3.24i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.81 + 3.14i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-9.07 - 2.43i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 0.815iT - 29T^{2} \)
31 \( 1 + (3.76 + 2.17i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.69 - 6.31i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 2.45iT - 41T^{2} \)
43 \( 1 + (3.59 - 3.59i)T - 43iT^{2} \)
47 \( 1 + (9.24 + 2.47i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.30 + 4.87i)T + (-45.8 + 26.5i)T^{2} \)
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 \( 1 + (15.3 - 4.10i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.68T + 71T^{2} \)
73 \( 1 + (-1.89 + 0.508i)T + (63.2 - 36.5i)T^{2} \)
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

−14.61323719007041269296380912738, −13.40217551265866739082646390144, −11.75585329067803323900688359558, −11.16534286562938909705197004532, −9.121670431107793877789139364780, −8.456062408216140944108717386270, −7.46074695033761143225537562993, −6.56213330025881968341220603487, −4.97661347870897894904430076015, −2.99357333005241772563751415647, 1.57878640858291730237383684010, 3.55620999852765921062475031024, 4.60587178749344161778496142921, 7.04397429959187457822540332212, 8.319835926699969714041383726567, 9.223898425974067538482883752796, 10.51645332923214982842893274995, 11.19912148273915436003588755993, 12.35837842030919188486971741064, 13.02010635872192766550725879999

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