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Properties

Label 2-945-63.5-c1-0-1
Degree $2$
Conductor $945$
Sign $-0.489 + 0.872i$
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  + 1.57i·2-s − 0.486·4-s + (0.5 + 0.866i)5-s + (−2.52 − 0.798i)7-s + 2.38i·8-s + (−1.36 + 0.788i)10-s + (−3.50 − 2.02i)11-s + (−2.45 − 1.41i)13-s + (1.25 − 3.97i)14-s − 4.73·16-s + (2.76 + 4.79i)17-s + (−7.37 − 4.25i)19-s + (−0.243 − 0.421i)20-s + (3.19 − 5.52i)22-s + (−0.825 + 0.476i)23-s + ⋯
L(s)  = 1  + 1.11i·2-s − 0.243·4-s + (0.223 + 0.387i)5-s + (−0.953 − 0.301i)7-s + 0.843i·8-s + (−0.431 + 0.249i)10-s + (−1.05 − 0.609i)11-s + (−0.680 − 0.392i)13-s + (0.336 − 1.06i)14-s − 1.18·16-s + (0.670 + 1.16i)17-s + (−1.69 − 0.976i)19-s + (−0.0543 − 0.0941i)20-s + (0.680 − 1.17i)22-s + (−0.172 + 0.0993i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.489 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $-0.489 + 0.872i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ -0.489 + 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.183789 - 0.313921i\)
\(L(\frac12)\) \(\approx\) \(0.183789 - 0.313921i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.52 + 0.798i)T \)
good2 \( 1 - 1.57iT - 2T^{2} \)
11 \( 1 + (3.50 + 2.02i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.45 + 1.41i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.76 - 4.79i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (7.37 + 4.25i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.825 - 0.476i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.77 - 1.60i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.36iT - 31T^{2} \)
37 \( 1 + (-1.10 + 1.90i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.62 - 2.80i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.84 - 4.93i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.33T + 47T^{2} \)
53 \( 1 + (-4.73 + 2.73i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 7.48T + 59T^{2} \)
61 \( 1 - 0.875iT - 61T^{2} \)
67 \( 1 + 7.80T + 67T^{2} \)
71 \( 1 - 9.34iT - 71T^{2} \)
73 \( 1 + (-14.1 + 8.18i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 7.01T + 79T^{2} \)
83 \( 1 + (2.79 + 4.83i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.18 + 2.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (14.7 - 8.51i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.61351394094081282409414628790, −9.754156195393227872776854576986, −8.645841660404694128611562682134, −7.941678819108267767781836714140, −7.13853196238967964103307205295, −6.32817798094036887706552079648, −5.78057829823990076148647229173, −4.73522898648748411691374572394, −3.30924056256525725631593412536, −2.32493455266581299155938487882, 0.14943400041682298392052052249, 1.99466876931094608971716133031, 2.68707100254126883256474783941, 3.83559938094246786012364878458, 4.87949715805096955532214996720, 5.98230861234603511495844381657, 6.92774336739050746545593065138, 7.84277041418542408826395706468, 9.044841825923434452523915400750, 9.826680275584577746239921717024

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