Properties

Label 2-105-15.8-c1-0-1
Degree $2$
Conductor $105$
Sign $-0.114 - 0.993i$
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  + (1.54 + 1.54i)2-s + (−1.73 − 0.00622i)3-s + 2.76i·4-s + (−0.252 + 2.22i)5-s + (−2.66 − 2.68i)6-s + (0.707 − 0.707i)7-s + (−1.18 + 1.18i)8-s + (2.99 + 0.0215i)9-s + (−3.82 + 3.04i)10-s − 3.38i·11-s + (0.0172 − 4.79i)12-s + (−0.206 − 0.206i)13-s + 2.18·14-s + (0.451 − 3.84i)15-s + 1.87·16-s + (0.167 + 0.167i)17-s + ⋯
L(s)  = 1  + (1.09 + 1.09i)2-s + (−0.999 − 0.00359i)3-s + 1.38i·4-s + (−0.112 + 0.993i)5-s + (−1.08 − 1.09i)6-s + (0.267 − 0.267i)7-s + (−0.419 + 0.419i)8-s + (0.999 + 0.00718i)9-s + (−1.20 + 0.961i)10-s − 1.02i·11-s + (0.00497 − 1.38i)12-s + (−0.0573 − 0.0573i)13-s + 0.583·14-s + (0.116 − 0.993i)15-s + 0.467·16-s + (0.0406 + 0.0406i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.114 - 0.993i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ -0.114 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.880395 + 0.988011i\)
\(L(\frac12)\) \(\approx\) \(0.880395 + 0.988011i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 + 0.00622i)T \)
5 \( 1 + (0.252 - 2.22i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good2 \( 1 + (-1.54 - 1.54i)T + 2iT^{2} \)
11 \( 1 + 3.38iT - 11T^{2} \)
13 \( 1 + (0.206 + 0.206i)T + 13iT^{2} \)
17 \( 1 + (-0.167 - 0.167i)T + 17iT^{2} \)
19 \( 1 + 5.31iT - 19T^{2} \)
23 \( 1 + (5.07 - 5.07i)T - 23iT^{2} \)
29 \( 1 + 2.84T + 29T^{2} \)
31 \( 1 - 9.11T + 31T^{2} \)
37 \( 1 + (5.27 - 5.27i)T - 37iT^{2} \)
41 \( 1 + 0.0314iT - 41T^{2} \)
43 \( 1 + (3.76 + 3.76i)T + 43iT^{2} \)
47 \( 1 + (3.56 + 3.56i)T + 47iT^{2} \)
53 \( 1 + (-3.55 + 3.55i)T - 53iT^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 + 6.80T + 61T^{2} \)
67 \( 1 + (-6.34 + 6.34i)T - 67iT^{2} \)
71 \( 1 + 3.95iT - 71T^{2} \)
73 \( 1 + (-8.61 - 8.61i)T + 73iT^{2} \)
79 \( 1 - 11.4iT - 79T^{2} \)
83 \( 1 + (-3.88 + 3.88i)T - 83iT^{2} \)
89 \( 1 - 2.00T + 89T^{2} \)
97 \( 1 + (-2.26 + 2.26i)T - 97iT^{2} \)
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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

−13.89879028617906143544059529504, −13.48638273580876918637618008159, −12.00488465243977598194206009864, −11.16948577699211139955045901898, −10.08067988533238116378434941847, −7.954300532562515027183071480760, −6.88774019054993463167942204129, −6.12156746897214066136202309806, −5.00136562283711947650359555900, −3.60927862754785861988965832633, 1.76119272322542106218804930515, 4.16339290432771893384852888771, 4.92644268719996325673704644914, 6.04930299926791623817787407931, 7.938077495866738493207703938437, 9.734408443410568030401969211854, 10.61664720473131664817320712014, 12.01015647408123916409667088874, 12.14355069735409468813365864228, 13.02069986590821447137324237705

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