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Properties

Label 2-945-63.20-c1-0-10
Degree $2$
Conductor $945$
Sign $-0.0155 - 0.999i$
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  + (1.5 + 0.866i)2-s + (0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 + 2.59i)7-s − 1.73i·8-s + 1.73i·10-s + (4.5 + 2.59i)11-s + (−4.5 + 2.59i)13-s + (−1.5 + 4.33i)14-s + (2.49 − 4.33i)16-s − 3·17-s + 3.46i·19-s + (−0.500 + 0.866i)20-s + (4.5 + 7.79i)22-s + (3 − 1.73i)23-s + ⋯
L(s)  = 1  + (1.06 + 0.612i)2-s + (0.250 + 0.433i)4-s + (0.223 + 0.387i)5-s + (0.188 + 0.981i)7-s − 0.612i·8-s + 0.547i·10-s + (1.35 + 0.783i)11-s + (−1.24 + 0.720i)13-s + (−0.400 + 1.15i)14-s + (0.624 − 1.08i)16-s − 0.727·17-s + 0.794i·19-s + (−0.111 + 0.193i)20-s + (0.959 + 1.66i)22-s + (0.625 − 0.361i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0155 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0155 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $-0.0155 - 0.999i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ -0.0155 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98357 + 2.01474i\)
\(L(\frac12)\) \(\approx\) \(1.98357 + 2.01474i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 2.59i)T \)
good2 \( 1 + (-1.5 - 0.866i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-4.5 - 2.59i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.5 - 2.59i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + (-3 + 1.73i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6 - 3.46i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6 + 3.46i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6.92iT - 53T^{2} \)
59 \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 + 1.73i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7 + 12.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.19iT - 71T^{2} \)
73 \( 1 + 8.66iT - 73T^{2} \)
79 \( 1 + (2.5 - 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (10.5 + 6.06i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.01806181453874433198673281585, −9.504022305800193155402344401174, −8.610192097001040514618764244668, −7.29823716532374335264735580833, −6.61765753052671418097686625290, −6.06238717532596700212653314991, −4.83419779239919406509778215170, −4.43247990686600628194650733169, −3.06757964457817207540539706905, −1.84665093772843274057267909480, 1.02171448310754620287544204420, 2.54195450176030097347197258894, 3.54809315598054458939993596823, 4.51354476249306609291933059130, 5.04149367818505917689109503501, 6.25089637806990588540837510908, 7.12937330138762303509354712690, 8.255228808762788063404788890301, 9.010867940313815811819819734787, 10.07865544429372724436402376266

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