Properties

Degree $2$
Conductor $978$
Sign $0.554 + 0.832i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.686 − 0.727i)2-s + (0.835 − 0.549i)3-s + (−0.0581 − 0.998i)4-s + (3.07 + 1.11i)5-s + (0.173 − 0.984i)6-s + (0.710 + 0.168i)7-s + (−0.766 − 0.642i)8-s + (0.396 − 0.918i)9-s + (2.92 − 1.46i)10-s + (1.45 − 0.955i)11-s + (−0.597 − 0.802i)12-s + (−1.35 + 1.13i)13-s + (0.610 − 0.401i)14-s + (3.17 − 0.753i)15-s + (−0.993 + 0.116i)16-s + (−2.07 − 1.74i)17-s + ⋯
L(s)  = 1  + (0.485 − 0.514i)2-s + (0.482 − 0.317i)3-s + (−0.0290 − 0.499i)4-s + (1.37 + 0.499i)5-s + (0.0708 − 0.402i)6-s + (0.268 + 0.0636i)7-s + (−0.270 − 0.227i)8-s + (0.132 − 0.306i)9-s + (0.923 − 0.463i)10-s + (0.438 − 0.288i)11-s + (−0.172 − 0.231i)12-s + (−0.375 + 0.314i)13-s + (0.163 − 0.107i)14-s + (0.821 − 0.194i)15-s + (−0.248 + 0.0290i)16-s + (−0.503 − 0.422i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 978 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 978 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(978\)    =    \(2 \cdot 3 \cdot 163\)
Sign: $0.554 + 0.832i$
Motivic weight: \(1\)
Character: $\chi_{978} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 978,\ (\ :1/2),\ 0.554 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.72729 - 1.46094i\)
\(L(\frac12)\) \(\approx\) \(2.72729 - 1.46094i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.686 + 0.727i)T \)
3 \( 1 + (-0.835 + 0.549i)T \)
163 \( 1 + (-8.53 - 9.49i)T \)
good5 \( 1 + (-3.07 - 1.11i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-0.710 - 0.168i)T + (6.25 + 3.14i)T^{2} \)
11 \( 1 + (-1.45 + 0.955i)T + (4.35 - 10.1i)T^{2} \)
13 \( 1 + (1.35 - 1.13i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (2.07 + 1.74i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (-1.01 + 3.39i)T + (-15.8 - 10.4i)T^{2} \)
23 \( 1 + (1.16 - 2.01i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.97 - 5.27i)T + (-1.68 + 28.9i)T^{2} \)
31 \( 1 + (-2.19 - 0.799i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (0.0843 + 0.478i)T + (-34.7 + 12.6i)T^{2} \)
41 \( 1 + (0.367 - 6.30i)T + (-40.7 - 4.75i)T^{2} \)
43 \( 1 + (3.74 + 1.88i)T + (25.6 + 34.4i)T^{2} \)
47 \( 1 + (-1.82 + 1.93i)T + (-2.73 - 46.9i)T^{2} \)
53 \( 1 + (-4.89 + 8.48i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 5.15T + 59T^{2} \)
61 \( 1 + (6.27 - 5.26i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (0.137 - 2.35i)T + (-66.5 - 7.77i)T^{2} \)
71 \( 1 + (5.44 + 2.73i)T + (42.3 + 56.9i)T^{2} \)
73 \( 1 + (-0.300 - 5.15i)T + (-72.5 + 8.47i)T^{2} \)
79 \( 1 + (-1.51 - 3.52i)T + (-54.2 + 57.4i)T^{2} \)
83 \( 1 + (-4.23 - 1.00i)T + (74.1 + 37.2i)T^{2} \)
89 \( 1 + (9.98 + 6.56i)T + (35.2 + 81.7i)T^{2} \)
97 \( 1 + (-4.33 - 10.0i)T + (-66.5 + 70.5i)T^{2} \)
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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

−9.835318354749389382624983792088, −9.270187419171896188231935483881, −8.412437482530036482630175908717, −7.01018815235280769053463899962, −6.51522010024880220540231925802, −5.48801405078836831518372940845, −4.59068214592532998080347683316, −3.19748839594517050440317617499, −2.40340173646527197300498141098, −1.43100271536998253888308031268, 1.67701554044286873804590337612, 2.74366338122886166992850618192, 4.13604221324642214761977067891, 4.88588308462178923277127555370, 5.84059239349891315870596487454, 6.49496818099657501042216208357, 7.69376487020931209107164023603, 8.470878353814511816956435288588, 9.296092818499767306611698349771, 9.950212414612684603213663220401

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