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Properties

Label 2-105-35.9-c1-0-7
Degree $2$
Conductor $105$
Sign $0.765 + 0.642i$
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  + (0.248 + 0.143i)2-s + (0.866 − 0.5i)3-s + (−0.958 − 1.66i)4-s + (0.717 − 2.11i)5-s + 0.287·6-s + (−1.11 + 2.39i)7-s − 1.12i·8-s + (0.499 − 0.866i)9-s + (0.482 − 0.423i)10-s + (1.66 + 2.88i)11-s + (−1.66 − 0.958i)12-s + 4.54i·13-s + (−0.622 + 0.436i)14-s + (−0.437 − 2.19i)15-s + (−1.75 + 3.04i)16-s + (4.80 − 2.77i)17-s + ⋯
L(s)  = 1  + (0.175 + 0.101i)2-s + (0.499 − 0.288i)3-s + (−0.479 − 0.830i)4-s + (0.321 − 0.947i)5-s + 0.117·6-s + (−0.421 + 0.906i)7-s − 0.397i·8-s + (0.166 − 0.288i)9-s + (0.152 − 0.134i)10-s + (0.502 + 0.869i)11-s + (−0.479 − 0.276i)12-s + 1.26i·13-s + (−0.166 + 0.116i)14-s + (−0.112 − 0.566i)15-s + (−0.438 + 0.760i)16-s + (1.16 − 0.672i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.765 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.765 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.765 + 0.642i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.765 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11316 - 0.405330i\)
\(L(\frac12)\) \(\approx\) \(1.11316 - 0.405330i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (-0.717 + 2.11i)T \)
7 \( 1 + (1.11 - 2.39i)T \)
good2 \( 1 + (-0.248 - 0.143i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-1.66 - 2.88i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.54iT - 13T^{2} \)
17 \( 1 + (-4.80 + 2.77i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.828 - 1.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.61 + 3.81i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.118T + 29T^{2} \)
31 \( 1 + (-3.13 - 5.42i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.71 + 3.87i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.0701T + 41T^{2} \)
43 \( 1 + 2.92iT - 43T^{2} \)
47 \( 1 + (-5.53 - 3.19i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.640 - 0.369i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.815 + 1.41i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.65 + 6.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.62 + 1.51i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.77T + 71T^{2} \)
73 \( 1 + (2.03 - 1.17i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.97 - 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.22iT - 83T^{2} \)
89 \( 1 + (6.50 - 11.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.04iT - 97T^{2} \)
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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

−13.96257708965066809539295329089, −12.51637872595430238306582469219, −12.08776062089114945868848619265, −9.978780848225778213728587238300, −9.359885577031086808324299475300, −8.495272859026153921811824018521, −6.70590232646612762267399572550, −5.53163681394575739975330124978, −4.26635399155184993567323056758, −1.83203030046010410540688880013, 3.12050933430970824980945030730, 3.85827958121493631181405105693, 5.87509730027442764817153476981, 7.44157859717153507168109974327, 8.291402155249058777236570936259, 9.773806151212839988620244540835, 10.53257686776302283969081934245, 11.83922687271231367966012688436, 13.19383108174390953068084547204, 13.78602010161964943936798044584

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