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Properties

Label 2-320-5.3-c2-0-19
Degree $2$
Conductor $320$
Sign $-0.995 + 0.0898i$
Analytic cond. $8.71936$
Root an. cond. $2.95285$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.64 − 2.64i)3-s + (−3 − 4i)5-s + (7.93 − 7.93i)7-s + 5.00i·9-s + 15.8·11-s + (−11 − 11i)13-s + (−2.64 + 18.5i)15-s + (−7 + 7i)17-s − 42.0·21-s + (−23.8 − 23.8i)23-s + (−7 + 24i)25-s + (−10.5 + 10.5i)27-s + 12i·29-s + 15.8·31-s + (−42.0 − 42.0i)33-s + ⋯
L(s)  = 1  + (−0.881 − 0.881i)3-s + (−0.600 − 0.800i)5-s + (1.13 − 1.13i)7-s + 0.555i·9-s + 1.44·11-s + (−0.846 − 0.846i)13-s + (−0.176 + 1.23i)15-s + (−0.411 + 0.411i)17-s − 2.00·21-s + (−1.03 − 1.03i)23-s + (−0.280 + 0.959i)25-s + (−0.391 + 0.391i)27-s + 0.413i·29-s + 0.512·31-s + (−1.27 − 1.27i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0898i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.995 + 0.0898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.995 + 0.0898i$
Analytic conductor: \(8.71936\)
Root analytic conductor: \(2.95285\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1),\ -0.995 + 0.0898i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0431768 - 0.959620i\)
\(L(\frac12)\) \(\approx\) \(0.0431768 - 0.959620i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (3 + 4i)T \)
good3 \( 1 + (2.64 + 2.64i)T + 9iT^{2} \)
7 \( 1 + (-7.93 + 7.93i)T - 49iT^{2} \)
11 \( 1 - 15.8T + 121T^{2} \)
13 \( 1 + (11 + 11i)T + 169iT^{2} \)
17 \( 1 + (7 - 7i)T - 289iT^{2} \)
19 \( 1 - 361T^{2} \)
23 \( 1 + (23.8 + 23.8i)T + 529iT^{2} \)
29 \( 1 - 12iT - 841T^{2} \)
31 \( 1 - 15.8T + 961T^{2} \)
37 \( 1 + (-5 + 5i)T - 1.36e3iT^{2} \)
41 \( 1 + 18T + 1.68e3T^{2} \)
43 \( 1 + (-7.93 - 7.93i)T + 1.84e3iT^{2} \)
47 \( 1 + (-7.93 + 7.93i)T - 2.20e3iT^{2} \)
53 \( 1 + (-53 - 53i)T + 2.80e3iT^{2} \)
59 \( 1 - 63.4iT - 3.48e3T^{2} \)
61 \( 1 + 78T + 3.72e3T^{2} \)
67 \( 1 + (-71.4 + 71.4i)T - 4.48e3iT^{2} \)
71 \( 1 - 15.8T + 5.04e3T^{2} \)
73 \( 1 + (47 + 47i)T + 5.32e3iT^{2} \)
79 \( 1 + 63.4iT - 6.24e3T^{2} \)
83 \( 1 + (55.5 + 55.5i)T + 6.88e3iT^{2} \)
89 \( 1 - 120iT - 7.92e3T^{2} \)
97 \( 1 + (-133 + 133i)T - 9.40e3iT^{2} \)
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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

−11.21365070211110486747613208707, −10.30353439229352893590568337571, −8.869887503723127161646188271198, −7.86884095927167663710533833303, −7.18386964422994289867468752259, −6.11495147939204924016674916761, −4.82375747616687413322608555189, −4.01896420935225311909906733262, −1.52669262511218942997903416032, −0.54293924574937295855552780528, 2.14122802780577868827166408955, 3.95342700801591079969308119258, 4.77612145449529687967925589374, 5.86002823819361356286986153946, 6.91173378917874943088717932931, 8.117105527072964701761369715550, 9.244821454442591229747384577034, 10.06564810374823646938281714636, 11.35576217540757750272172769503, 11.59469428442379697648253950689

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