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Properties

Label 2-105-35.13-c1-0-6
Degree $2$
Conductor $105$
Sign $0.443 + 0.896i$
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  + (1.48 − 1.48i)2-s + (−0.707 + 0.707i)3-s − 2.43i·4-s + (1.28 − 1.82i)5-s + 2.10i·6-s + (−1.75 + 1.97i)7-s + (−0.640 − 0.640i)8-s − 1.00i·9-s + (−0.798 − 4.63i)10-s − 2.67·11-s + (1.71 + 1.71i)12-s + (−1.22 + 1.22i)13-s + (0.320 + 5.55i)14-s + (0.379 + 2.20i)15-s + 2.95·16-s + (4.74 + 4.74i)17-s + ⋯
L(s)  = 1  + (1.05 − 1.05i)2-s + (−0.408 + 0.408i)3-s − 1.21i·4-s + (0.576 − 0.816i)5-s + 0.859i·6-s + (−0.665 + 0.746i)7-s + (−0.226 − 0.226i)8-s − 0.333i·9-s + (−0.252 − 1.46i)10-s − 0.805·11-s + (0.496 + 0.496i)12-s + (−0.340 + 0.340i)13-s + (0.0857 + 1.48i)14-s + (0.0979 + 0.568i)15-s + 0.738·16-s + (1.15 + 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.443 + 0.896i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.443 + 0.896i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26862 - 0.787305i\)
\(L(\frac12)\) \(\approx\) \(1.26862 - 0.787305i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-1.28 + 1.82i)T \)
7 \( 1 + (1.75 - 1.97i)T \)
good2 \( 1 + (-1.48 + 1.48i)T - 2iT^{2} \)
11 \( 1 + 2.67T + 11T^{2} \)
13 \( 1 + (1.22 - 1.22i)T - 13iT^{2} \)
17 \( 1 + (-4.74 - 4.74i)T + 17iT^{2} \)
19 \( 1 + 6.01T + 19T^{2} \)
23 \( 1 + (0.175 + 0.175i)T + 23iT^{2} \)
29 \( 1 - 0.304iT - 29T^{2} \)
31 \( 1 + 7.25iT - 31T^{2} \)
37 \( 1 + (0.735 - 0.735i)T - 37iT^{2} \)
41 \( 1 + 7.05iT - 41T^{2} \)
43 \( 1 + (-0.304 - 0.304i)T + 43iT^{2} \)
47 \( 1 + (0.556 + 0.556i)T + 47iT^{2} \)
53 \( 1 + (4.99 + 4.99i)T + 53iT^{2} \)
59 \( 1 - 7.98T + 59T^{2} \)
61 \( 1 - 5.53iT - 61T^{2} \)
67 \( 1 + (3.43 - 3.43i)T - 67iT^{2} \)
71 \( 1 - 15.3T + 71T^{2} \)
73 \( 1 + (10.0 - 10.0i)T - 73iT^{2} \)
79 \( 1 - 11.2iT - 79T^{2} \)
83 \( 1 + (-4.88 + 4.88i)T - 83iT^{2} \)
89 \( 1 + 6.91T + 89T^{2} \)
97 \( 1 + (-8.84 - 8.84i)T + 97iT^{2} \)
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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

−13.02825583424224659807547311040, −12.70204894180904857698640093866, −11.80065339443552993838825825667, −10.50253493954416741366167364304, −9.769631663070175619063329750045, −8.382089878248136318605933992594, −6.03014666693700717711792218993, −5.25306099255159287900039688562, −3.96351246892718436237204515860, −2.26675257482203083097242262209, 3.14774952490807975471958947124, 4.94469544713314655053931170402, 6.07544782999447169228396953131, 6.94449627271380016053651299234, 7.75473144207418836566183568892, 9.926152481612053452362094039768, 10.74280493177525055304710924216, 12.39985170499961092071568700970, 13.20461508243351978949758959535, 13.96575362278167076398685625496

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