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Properties

Label 2-320-5.2-c4-0-25
Degree $2$
Conductor $320$
Sign $0.365 - 0.930i$
Analytic cond. $33.0783$
Root an. cond. $5.75138$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − i)3-s + (15 + 20i)5-s + (19 + 19i)7-s + 79i·9-s + 202·11-s + (99 − 99i)13-s + (35 + 5i)15-s + (−239 − 239i)17-s + 40i·19-s + 38·21-s + (−541 + 541i)23-s + (−175 + 600i)25-s + (160 + 160i)27-s + 200i·29-s + 758·31-s + ⋯
L(s)  = 1  + (0.111 − 0.111i)3-s + (0.599 + 0.800i)5-s + (0.387 + 0.387i)7-s + 0.975i·9-s + 1.66·11-s + (0.585 − 0.585i)13-s + (0.155 + 0.0222i)15-s + (−0.826 − 0.826i)17-s + 0.110i·19-s + 0.0861·21-s + (−1.02 + 1.02i)23-s + (−0.280 + 0.960i)25-s + (0.219 + 0.219i)27-s + 0.237i·29-s + 0.788·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.365 - 0.930i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.365 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.365 - 0.930i$
Analytic conductor: \(33.0783\)
Root analytic conductor: \(5.75138\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :2),\ 0.365 - 0.930i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.590458872\)
\(L(\frac12)\) \(\approx\) \(2.590458872\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-15 - 20i)T \)
good3 \( 1 + (-1 + i)T - 81iT^{2} \)
7 \( 1 + (-19 - 19i)T + 2.40e3iT^{2} \)
11 \( 1 - 202T + 1.46e4T^{2} \)
13 \( 1 + (-99 + 99i)T - 2.85e4iT^{2} \)
17 \( 1 + (239 + 239i)T + 8.35e4iT^{2} \)
19 \( 1 - 40iT - 1.30e5T^{2} \)
23 \( 1 + (541 - 541i)T - 2.79e5iT^{2} \)
29 \( 1 - 200iT - 7.07e5T^{2} \)
31 \( 1 - 758T + 9.23e5T^{2} \)
37 \( 1 + (141 + 141i)T + 1.87e6iT^{2} \)
41 \( 1 - 1.04e3T + 2.82e6T^{2} \)
43 \( 1 + (759 - 759i)T - 3.41e6iT^{2} \)
47 \( 1 + (-459 - 459i)T + 4.87e6iT^{2} \)
53 \( 1 + (-1.81e3 + 1.81e3i)T - 7.89e6iT^{2} \)
59 \( 1 - 4.60e3iT - 1.21e7T^{2} \)
61 \( 1 + 2.08e3T + 1.38e7T^{2} \)
67 \( 1 + (-5.08e3 - 5.08e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 3.47e3T + 2.54e7T^{2} \)
73 \( 1 + (3.47e3 - 3.47e3i)T - 2.83e7iT^{2} \)
79 \( 1 - 7.68e3iT - 3.89e7T^{2} \)
83 \( 1 + (-6.08e3 + 6.08e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 5.68e3iT - 6.27e7T^{2} \)
97 \( 1 + (-561 - 561i)T + 8.85e7iT^{2} \)
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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

−11.27391601029339718807031535637, −10.25751425940162847092159798646, −9.331113397090419715387663425093, −8.372404772905951039911123392705, −7.25634501747331978266938684473, −6.31942785759780818929952310854, −5.34483386515795312367520154484, −3.93521824054533975124336756754, −2.55746946151074549255495552373, −1.47111986687272980984409109576, 0.834160949294571634865814982994, 1.87913766722440240789953527796, 3.88085207395111312540979035209, 4.46969548623944088944857988989, 6.17751622690732972445204702457, 6.57247613543210199912155824064, 8.286020009103480698859821383525, 8.989637200837712035739620657488, 9.657579075164835445887825028667, 10.82014394470428063134006263262

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