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Properties

Label 2-945-15.2-c1-0-37
Degree $2$
Conductor $945$
Sign $0.240 + 0.970i$
Analytic cond. $7.54586$
Root an. cond. $2.74697$
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  + (0.885 − 0.885i)2-s + 0.430i·4-s + (0.684 − 2.12i)5-s + (−0.707 − 0.707i)7-s + (2.15 + 2.15i)8-s + (−1.28 − 2.49i)10-s − 0.102i·11-s + (−0.320 + 0.320i)13-s − 1.25·14-s + 2.95·16-s + (4.37 − 4.37i)17-s − 6.18i·19-s + (0.915 + 0.294i)20-s + (−0.0910 − 0.0910i)22-s + (1.19 + 1.19i)23-s + ⋯
L(s)  = 1  + (0.626 − 0.626i)2-s + 0.215i·4-s + (0.305 − 0.952i)5-s + (−0.267 − 0.267i)7-s + (0.761 + 0.761i)8-s + (−0.404 − 0.788i)10-s − 0.0309i·11-s + (−0.0889 + 0.0889i)13-s − 0.334·14-s + 0.738·16-s + (1.06 − 1.06i)17-s − 1.41i·19-s + (0.204 + 0.0657i)20-s + (−0.0194 − 0.0194i)22-s + (0.250 + 0.250i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.240 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.240 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.240 + 0.970i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (512, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ 0.240 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86896 - 1.46272i\)
\(L(\frac12)\) \(\approx\) \(1.86896 - 1.46272i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-0.684 + 2.12i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (-0.885 + 0.885i)T - 2iT^{2} \)
11 \( 1 + 0.102iT - 11T^{2} \)
13 \( 1 + (0.320 - 0.320i)T - 13iT^{2} \)
17 \( 1 + (-4.37 + 4.37i)T - 17iT^{2} \)
19 \( 1 + 6.18iT - 19T^{2} \)
23 \( 1 + (-1.19 - 1.19i)T + 23iT^{2} \)
29 \( 1 - 6.40T + 29T^{2} \)
31 \( 1 - 6.61T + 31T^{2} \)
37 \( 1 + (3.33 + 3.33i)T + 37iT^{2} \)
41 \( 1 - 6.60iT - 41T^{2} \)
43 \( 1 + (2.74 - 2.74i)T - 43iT^{2} \)
47 \( 1 + (-1.77 + 1.77i)T - 47iT^{2} \)
53 \( 1 + (2.88 + 2.88i)T + 53iT^{2} \)
59 \( 1 + 12.8T + 59T^{2} \)
61 \( 1 + 3.25T + 61T^{2} \)
67 \( 1 + (1.55 + 1.55i)T + 67iT^{2} \)
71 \( 1 - 3.88iT - 71T^{2} \)
73 \( 1 + (-9.24 + 9.24i)T - 73iT^{2} \)
79 \( 1 + 2.10iT - 79T^{2} \)
83 \( 1 + (-8.52 - 8.52i)T + 83iT^{2} \)
89 \( 1 - 12.6T + 89T^{2} \)
97 \( 1 + (10.4 + 10.4i)T + 97iT^{2} \)
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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

−9.875614278901702727146493129107, −9.153395711766112424264784107713, −8.220574438531316921710476482245, −7.43285602083559270740985451235, −6.32574175550191083300220515121, −4.95317686221612855518284862638, −4.75090042706916596436140131643, −3.41021852191267260542487523586, −2.52835709583468890917683692952, −1.02769569413763797266878605592, 1.57102654195378158516399409209, 3.05551339708698419052781890487, 4.00472701860206307134693624492, 5.21958272520217251748881260568, 6.07462177424837157893131908519, 6.47991116573788491606557625077, 7.50679190056572830574556501359, 8.310540799370700521758459738611, 9.656768846953831429428616069767, 10.29514330238665465938638201371

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