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Properties

Label 2-320-320.29-c1-0-10
Degree $2$
Conductor $320$
Sign $0.520 - 0.853i$
Analytic cond. $2.55521$
Root an. cond. $1.59850$
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  + (0.635 − 1.26i)2-s + (−0.627 + 3.15i)3-s + (−1.19 − 1.60i)4-s + (2.05 + 0.873i)5-s + (3.58 + 2.79i)6-s + (0.393 + 0.950i)7-s + (−2.78 + 0.485i)8-s + (−6.79 − 2.81i)9-s + (2.41 − 2.04i)10-s + (0.793 + 3.98i)11-s + (5.81 − 2.75i)12-s + (3.25 − 2.17i)13-s + (1.45 + 0.106i)14-s + (−4.05 + 5.94i)15-s + (−1.15 + 3.82i)16-s + (−4.57 + 4.57i)17-s + ⋯
L(s)  = 1  + (0.449 − 0.893i)2-s + (−0.362 + 1.82i)3-s + (−0.596 − 0.802i)4-s + (0.920 + 0.390i)5-s + (1.46 + 1.14i)6-s + (0.148 + 0.359i)7-s + (−0.985 + 0.171i)8-s + (−2.26 − 0.938i)9-s + (0.762 − 0.646i)10-s + (0.239 + 1.20i)11-s + (1.67 − 0.794i)12-s + (0.903 − 0.603i)13-s + (0.387 + 0.0285i)14-s + (−1.04 + 1.53i)15-s + (−0.289 + 0.957i)16-s + (−1.10 + 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.520 - 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.520 - 0.853i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.520 - 0.853i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29565 + 0.727459i\)
\(L(\frac12)\) \(\approx\) \(1.29565 + 0.727459i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.635 + 1.26i)T \)
5 \( 1 + (-2.05 - 0.873i)T \)
good3 \( 1 + (0.627 - 3.15i)T + (-2.77 - 1.14i)T^{2} \)
7 \( 1 + (-0.393 - 0.950i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-0.793 - 3.98i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 + (-3.25 + 2.17i)T + (4.97 - 12.0i)T^{2} \)
17 \( 1 + (4.57 - 4.57i)T - 17iT^{2} \)
19 \( 1 + (1.03 - 0.690i)T + (7.27 - 17.5i)T^{2} \)
23 \( 1 + (-4.62 - 1.91i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-1.04 + 5.23i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + 5.14iT - 31T^{2} \)
37 \( 1 + (-0.223 + 0.334i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (-3.90 + 9.42i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (-0.307 - 1.54i)T + (-39.7 + 16.4i)T^{2} \)
47 \( 1 + (2.52 - 2.52i)T - 47iT^{2} \)
53 \( 1 + (-3.10 + 0.617i)T + (48.9 - 20.2i)T^{2} \)
59 \( 1 + (-3.05 + 4.57i)T + (-22.5 - 54.5i)T^{2} \)
61 \( 1 + (6.06 + 1.20i)T + (56.3 + 23.3i)T^{2} \)
67 \( 1 + (-2.49 + 12.5i)T + (-61.8 - 25.6i)T^{2} \)
71 \( 1 + (1.29 - 0.536i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (1.82 - 4.41i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (5.76 - 5.76i)T - 79iT^{2} \)
83 \( 1 + (-1.79 - 2.68i)T + (-31.7 + 76.6i)T^{2} \)
89 \( 1 + (1.53 + 3.70i)T + (-62.9 + 62.9i)T^{2} \)
97 \( 1 - 1.37T + 97T^{2} \)
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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

−11.36419534128133414502529100759, −10.79246934919530641028430170491, −10.12814984388622770423330515451, −9.400321156690825139953798842938, −8.692899193084594946708031191031, −6.28411753772269850198852141829, −5.55574774850047987461243718443, −4.54687902985626881730806373042, −3.65979821256775258155578088471, −2.26828032962958419817343494368, 1.05675502414230779933490429431, 2.81734419812555733163870924888, 4.84707642590102927914343646225, 5.94121003782626874747263564937, 6.57104404425477014172406216645, 7.25064807974261444580506909315, 8.678286330102834206143509725253, 8.833018135305304874760964972556, 10.97111203818761283837440910378, 11.72662359263556110048726855707

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