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Properties

Label 2-105-35.3-c1-0-5
Degree $2$
Conductor $105$
Sign $0.820 + 0.570i$
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  + (0.648 + 0.173i)2-s + (−0.258 − 0.965i)3-s + (−1.34 − 0.774i)4-s + (2.13 − 0.672i)5-s − 0.671i·6-s + (2.57 + 0.588i)7-s + (−1.68 − 1.68i)8-s + (−0.866 + 0.499i)9-s + (1.49 − 0.0654i)10-s + (0.0701 − 0.121i)11-s + (−0.401 + 1.49i)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 + (−7.37 + 1.97i)17-s + ⋯
L(s)  = 1  + (0.458 + 0.122i)2-s + (−0.149 − 0.557i)3-s + (−0.670 − 0.387i)4-s + (0.953 − 0.300i)5-s − 0.273i·6-s + (0.974 + 0.222i)7-s + (−0.595 − 0.595i)8-s + (−0.288 + 0.166i)9-s + (0.474 − 0.0206i)10-s + (0.0211 − 0.0366i)11-s + (−0.115 + 0.432i)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 + (−1.78 + 0.478i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.570i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.820 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.820 + 0.570i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.820 + 0.570i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15361 - 0.361694i\)
\(L(\frac12)\) \(\approx\) \(1.15361 - 0.361694i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (-2.13 + 0.672i)T \)
7 \( 1 + (-2.57 - 0.588i)T \)
good2 \( 1 + (-0.648 - 0.173i)T + (1.73 + i)T^{2} \)
11 \( 1 + (-0.0701 + 0.121i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.35 - 2.35i)T - 13iT^{2} \)
17 \( 1 + (7.37 - 1.97i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-3.89 - 6.74i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.671 + 2.50i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 5.09iT - 29T^{2} \)
31 \( 1 + (-2.54 - 1.46i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.73 + 1.53i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 0.261iT - 41T^{2} \)
43 \( 1 + (2.11 + 2.11i)T + 43iT^{2} \)
47 \( 1 + (-0.402 + 1.50i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.79 - 0.749i)T + (45.8 - 26.5i)T^{2} \)
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 \( 1 + (2.02 + 7.54i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.56T + 71T^{2} \)
73 \( 1 + (0.847 + 3.16i)T + (-63.2 + 36.5i)T^{2} \)
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

−13.85820106457372730589693413769, −12.80201707479960089949674056874, −11.86820734002845865418160478967, −10.44963486312922793539058696044, −9.281116949554633966426032765099, −8.303934176551326565175209384189, −6.58915552201595879844574855508, −5.51963814123151431955341821532, −4.48930740899468620161281477656, −1.86559611621329182738348433253, 2.80060338080832838071840590703, 4.65060148068692298429087978080, 5.31271514452969935626734399400, 7.10295483558566948955476127369, 8.695257687099484546345898754841, 9.533726834784980364483133956550, 10.81510247648722732998645783454, 11.69935558715757393713179363638, 13.15432403455143165516775640958, 13.76963416307517355107924317775

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