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Properties

Label 2-945-315.59-c1-0-14
Degree $2$
Conductor $945$
Sign $-0.738 - 0.674i$
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  + (−0.466 + 0.807i)2-s + (0.565 + 0.979i)4-s + (1.03 + 1.98i)5-s + (2.46 + 0.955i)7-s − 2.91·8-s + (−2.08 − 0.0907i)10-s − 1.22i·11-s + (−1.31 + 2.28i)13-s + (−1.92 + 1.54i)14-s + (0.229 − 0.397i)16-s + (6.76 + 3.90i)17-s + (2.02 − 1.16i)19-s + (−1.35 + 2.13i)20-s + (0.986 + 0.569i)22-s − 6.98·23-s + ⋯
L(s)  = 1  + (−0.329 + 0.570i)2-s + (0.282 + 0.489i)4-s + (0.461 + 0.886i)5-s + (0.932 + 0.361i)7-s − 1.03·8-s + (−0.658 − 0.0286i)10-s − 0.368i·11-s + (−0.365 + 0.633i)13-s + (−0.513 + 0.413i)14-s + (0.0573 − 0.0993i)16-s + (1.64 + 0.947i)17-s + (0.463 − 0.267i)19-s + (−0.303 + 0.476i)20-s + (0.210 + 0.121i)22-s − 1.45·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.738 - 0.674i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.738 - 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $-0.738 - 0.674i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (584, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ -0.738 - 0.674i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.585264 + 1.50794i\)
\(L(\frac12)\) \(\approx\) \(0.585264 + 1.50794i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.03 - 1.98i)T \)
7 \( 1 + (-2.46 - 0.955i)T \)
good2 \( 1 + (0.466 - 0.807i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + 1.22iT - 11T^{2} \)
13 \( 1 + (1.31 - 2.28i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-6.76 - 3.90i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.02 + 1.16i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 6.98T + 23T^{2} \)
29 \( 1 + (-5.69 + 3.28i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.58 - 2.06i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.395 - 0.228i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.67 + 4.63i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.97 + 4.02i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.96 + 2.28i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.18 - 2.04i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.41 + 5.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.253 + 0.146i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.7 - 7.38i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.44iT - 71T^{2} \)
73 \( 1 + (-3.51 + 6.09i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.75 - 9.96i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.79 - 1.03i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.840 - 1.45i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.93 + 5.09i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.30260309280984415864203958408, −9.461037283044162439698366712868, −8.461819011392564706447492214532, −7.80999116757703730373344165712, −7.13437347750325853011270981163, −6.08944208808001690662668795001, −5.56093439929966433420545115274, −4.01138658053281565671777076528, −2.94219058955963527758287247768, −1.86540018062896275879049435023, 0.878571681998174204845782696668, 1.76257604166693382834687697315, 3.00513702766456333390855312033, 4.52924410180181842306517904253, 5.36616404148534877061096044232, 6.01443611561907855615919567107, 7.46935453068766873113294758799, 8.059479282175538571482008818784, 9.161359056008371236985076354436, 9.907119078198065672649937488240

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