Properties

Label 2-105-35.17-c1-0-2
Degree $2$
Conductor $105$
Sign $0.593 - 0.805i$
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  + (−0.173 + 0.648i)2-s + (−0.965 + 0.258i)3-s + (1.34 + 0.774i)4-s + (1.64 − 1.51i)5-s − 0.671i·6-s + (−0.588 + 2.57i)7-s + (−1.68 + 1.68i)8-s + (0.866 − 0.499i)9-s + (0.692 + 1.33i)10-s + (0.0701 − 0.121i)11-s + (−1.49 − 0.401i)12-s + (2.35 + 2.35i)13-s + (−1.56 − 0.829i)14-s + (−1.20 + 1.88i)15-s + (0.750 + 1.29i)16-s + (−1.97 − 7.37i)17-s + ⋯
L(s)  = 1  + (−0.122 + 0.458i)2-s + (−0.557 + 0.149i)3-s + (0.670 + 0.387i)4-s + (0.737 − 0.675i)5-s − 0.273i·6-s + (−0.222 + 0.974i)7-s + (−0.595 + 0.595i)8-s + (0.288 − 0.166i)9-s + (0.219 + 0.420i)10-s + (0.0211 − 0.0366i)11-s + (−0.432 − 0.115i)12-s + (0.653 + 0.653i)13-s + (−0.419 − 0.221i)14-s + (−0.310 + 0.486i)15-s + (0.187 + 0.324i)16-s + (−0.478 − 1.78i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.593 - 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.593 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.593 - 0.805i$
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: \((2,\ 105,\ (\ :1/2),\ 0.593 - 0.805i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.900522 + 0.455097i\)
\(L(\frac12)\) \(\approx\) \(0.900522 + 0.455097i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 + 1.51i)T \)
7 \( 1 + (0.588 - 2.57i)T \)
good2 \( 1 + (0.173 - 0.648i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (-0.0701 + 0.121i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.35 - 2.35i)T + 13iT^{2} \)
17 \( 1 + (1.97 + 7.37i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.89 + 6.74i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.50 + 0.671i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 5.09iT - 29T^{2} \)
31 \( 1 + (-2.54 - 1.46i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.53 + 5.73i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 0.261iT - 41T^{2} \)
43 \( 1 + (2.11 - 2.11i)T - 43iT^{2} \)
47 \( 1 + (-1.50 - 0.402i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.749 - 2.79i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (4.37 - 7.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.76 - 2.75i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.54 + 2.02i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 3.56T + 71T^{2} \)
73 \( 1 + (3.16 - 0.847i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-0.113 + 0.0656i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.33 - 7.33i)T + 83iT^{2} \)
89 \( 1 + (-2.44 - 4.23i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.25 - 1.25i)T - 97iT^{2} \)
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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

−13.88197336755124518111783728562, −12.74652269762622891978078120082, −11.84003148721363024422273001534, −10.97748799101672487247212209108, −9.306495631829939313424785463888, −8.697400931834297854101352642169, −6.93216311914812382519774127263, −6.06319266035564652210869583541, −4.87444142054559806792466160889, −2.46946980695583669959464340699, 1.75634072181531211625844970215, 3.71896196892105431101759147812, 6.04648488148743836958452327119, 6.43227041572364140228587234974, 7.976029602748170883013901283474, 10.03508101836420162353250107519, 10.40390146314393960234259150461, 11.20080855119709782923709793481, 12.54035967849281763276145737992, 13.44436787601062735420107333516

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