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Properties

Label 2-105-105.74-c2-0-25
Degree $2$
Conductor $105$
Sign $-0.978 + 0.205i$
Analytic cond. $2.86104$
Root an. cond. $1.69146$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.530 − 0.919i)2-s + (−1.89 − 2.32i)3-s + (1.43 − 2.48i)4-s + (−0.901 − 4.91i)5-s + (−1.12 + 2.97i)6-s + (3.04 + 6.30i)7-s − 7.29·8-s + (−1.78 + 8.82i)9-s + (−4.04 + 3.43i)10-s + (−9.99 − 5.76i)11-s + (−8.50 + 1.38i)12-s − 5.56i·13-s + (4.17 − 6.14i)14-s + (−9.70 + 11.4i)15-s + (−1.87 − 3.24i)16-s + (7.82 − 13.5i)17-s + ⋯
L(s)  = 1  + (−0.265 − 0.459i)2-s + (−0.633 − 0.774i)3-s + (0.359 − 0.622i)4-s + (−0.180 − 0.983i)5-s + (−0.187 + 0.496i)6-s + (0.435 + 0.900i)7-s − 0.912·8-s + (−0.198 + 0.980i)9-s + (−0.404 + 0.343i)10-s + (−0.908 − 0.524i)11-s + (−0.708 + 0.115i)12-s − 0.428i·13-s + (0.298 − 0.439i)14-s + (−0.647 + 0.762i)15-s + (−0.117 − 0.202i)16-s + (0.460 − 0.797i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.978 + 0.205i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ -0.978 + 0.205i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0866680 - 0.836191i\)
\(L(\frac12)\) \(\approx\) \(0.0866680 - 0.836191i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.89 + 2.32i)T \)
5 \( 1 + (0.901 + 4.91i)T \)
7 \( 1 + (-3.04 - 6.30i)T \)
good2 \( 1 + (0.530 + 0.919i)T + (-2 + 3.46i)T^{2} \)
11 \( 1 + (9.99 + 5.76i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 5.56iT - 169T^{2} \)
17 \( 1 + (-7.82 + 13.5i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-1.06 - 1.84i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-13.0 - 22.5i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 13.3iT - 841T^{2} \)
31 \( 1 + (-25.3 + 43.9i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-19.8 + 11.4i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 53.9iT - 1.68e3T^{2} \)
43 \( 1 + 38.8iT - 1.84e3T^{2} \)
47 \( 1 + (-2.72 - 4.71i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-15.5 + 26.8i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-97.9 - 56.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (14.8 + 25.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (23.5 + 13.6i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 79.6iT - 5.04e3T^{2} \)
73 \( 1 + (1.70 + 0.981i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-43.8 - 75.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 71.3T + 6.88e3T^{2} \)
89 \( 1 + (106. - 61.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 1.02iT - 9.40e3T^{2} \)
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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

−12.77351771987372641513906068705, −11.79810998289235855083692166131, −11.27579618155883065122624219972, −9.926395844275869713160444256223, −8.652278555509174814751863966716, −7.56221770698715395506484219205, −5.74945110533179800718143638109, −5.28545959070592479532940261541, −2.37525987648620775742984734512, −0.73295629672955769027688904652, 3.15789989213865363061611904673, 4.57604767841661733329410062351, 6.34456871111593429967385413817, 7.23549341094272282220702658125, 8.359671971759394315902958347418, 10.03434086809809346945417870514, 10.77336552881020736414244053663, 11.65669330262901645573390673266, 12.81562549890636148630459233433, 14.43712974584529651728306993023

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