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Properties

Label 2-945-945.734-c1-0-31
Degree $2$
Conductor $945$
Sign $0.734 - 0.678i$
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  + (0.306 − 1.70i)3-s + (−0.347 + 1.96i)4-s + (−0.764 − 2.10i)5-s + (0.459 + 2.60i)7-s + (−2.81 − 1.04i)9-s + (−0.890 + 2.44i)11-s + (3.25 + 1.19i)12-s + (0.146 + 0.123i)13-s + (−3.81 + 0.659i)15-s + (−3.75 − 1.36i)16-s + (7.10 + 4.10i)17-s + (4.40 − 0.776i)20-s + (4.58 + 0.0162i)21-s + (−3.83 + 3.21i)25-s + (−2.64 + 4.47i)27-s − 5.29·28-s + ⋯
L(s)  = 1  + (0.177 − 0.984i)3-s + (−0.173 + 0.984i)4-s + (−0.342 − 0.939i)5-s + (0.173 + 0.984i)7-s + (−0.937 − 0.348i)9-s + (−0.268 + 0.737i)11-s + (0.938 + 0.345i)12-s + (0.0407 + 0.0342i)13-s + (−0.985 + 0.170i)15-s + (−0.939 − 0.342i)16-s + (1.72 + 0.994i)17-s + (0.984 − 0.173i)20-s + (0.999 + 0.00354i)21-s + (−0.766 + 0.642i)25-s + (−0.509 + 0.860i)27-s − 0.999·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.734 - 0.678i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (734, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ 0.734 - 0.678i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22384 + 0.478688i\)
\(L(\frac12)\) \(\approx\) \(1.22384 + 0.478688i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.306 + 1.70i)T \)
5 \( 1 + (0.764 + 2.10i)T \)
7 \( 1 + (-0.459 - 2.60i)T \)
good2 \( 1 + (0.347 - 1.96i)T^{2} \)
11 \( 1 + (0.890 - 2.44i)T + (-8.42 - 7.07i)T^{2} \)
13 \( 1 + (-0.146 - 0.123i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-7.10 - 4.10i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-6.91 - 8.23i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (29.1 + 10.6i)T^{2} \)
37 \( 1 + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (-7.67 + 1.35i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (-8.45 - 4.88i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.52 - 13.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (9.32 - 7.82i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (10.9 + 13.0i)T + (-14.4 + 81.7i)T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.18 + 1.15i)T + (74.3 + 62.3i)T^{2} \)
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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

−9.943163005882543662267579426926, −8.872449321871828120708699462125, −8.423455490030555324242312330791, −7.80651954990197269581772459793, −7.01210446233736205686935138991, −5.76985885435924821091843698856, −4.96491480566547594985721928611, −3.70241550939343676668355028119, −2.65756639954875154034744528038, −1.39251828898223199893123588991, 0.65989268551297406307656107220, 2.69641430363562963309654444037, 3.64736699787875550671684417098, 4.58919865838233162769631958360, 5.53804153322224873373888244078, 6.37089411877486234648458504796, 7.52243078902674413109746472360, 8.242714625773982217580042328640, 9.483827517868334796985723966913, 10.08780540570416048287407931024

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