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Properties

Label 2-1360-20.3-c1-0-4
Degree $2$
Conductor $1360$
Sign $-0.569 + 0.821i$
Analytic cond. $10.8596$
Root an. cond. $3.29539$
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  + (−2.19 + 2.19i)3-s + (1.49 + 1.66i)5-s + (2.96 + 2.96i)7-s − 6.60i·9-s − 1.38i·11-s + (−4.45 − 4.45i)13-s + (−6.91 − 0.364i)15-s + (−0.707 + 0.707i)17-s − 4.88·19-s − 12.9·21-s + (−3.78 + 3.78i)23-s + (−0.525 + 4.97i)25-s + (7.89 + 7.89i)27-s − 6.11i·29-s − 3.24i·31-s + ⋯
L(s)  = 1  + (−1.26 + 1.26i)3-s + (0.668 + 0.743i)5-s + (1.11 + 1.11i)7-s − 2.20i·9-s − 0.416i·11-s + (−1.23 − 1.23i)13-s + (−1.78 − 0.0942i)15-s + (−0.171 + 0.171i)17-s − 1.12·19-s − 2.83·21-s + (−0.789 + 0.789i)23-s + (−0.105 + 0.994i)25-s + (1.51 + 1.51i)27-s − 1.13i·29-s − 0.583i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.569 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1360\)    =    \(2^{4} \cdot 5 \cdot 17\)
Sign: $-0.569 + 0.821i$
Analytic conductor: \(10.8596\)
Root analytic conductor: \(3.29539\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1360} (783, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1360,\ (\ :1/2),\ -0.569 + 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3713017393\)
\(L(\frac12)\) \(\approx\) \(0.3713017393\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-1.49 - 1.66i)T \)
17 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (2.19 - 2.19i)T - 3iT^{2} \)
7 \( 1 + (-2.96 - 2.96i)T + 7iT^{2} \)
11 \( 1 + 1.38iT - 11T^{2} \)
13 \( 1 + (4.45 + 4.45i)T + 13iT^{2} \)
19 \( 1 + 4.88T + 19T^{2} \)
23 \( 1 + (3.78 - 3.78i)T - 23iT^{2} \)
29 \( 1 + 6.11iT - 29T^{2} \)
31 \( 1 + 3.24iT - 31T^{2} \)
37 \( 1 + (3.88 - 3.88i)T - 37iT^{2} \)
41 \( 1 + 7.98T + 41T^{2} \)
43 \( 1 + (3.84 - 3.84i)T - 43iT^{2} \)
47 \( 1 + (-2.22 - 2.22i)T + 47iT^{2} \)
53 \( 1 + (4.67 + 4.67i)T + 53iT^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 - 3.02T + 61T^{2} \)
67 \( 1 + (5.47 + 5.47i)T + 67iT^{2} \)
71 \( 1 - 1.86iT - 71T^{2} \)
73 \( 1 + (-5.76 - 5.76i)T + 73iT^{2} \)
79 \( 1 + 17.0T + 79T^{2} \)
83 \( 1 + (2.65 - 2.65i)T - 83iT^{2} \)
89 \( 1 - 11.1iT - 89T^{2} \)
97 \( 1 + (-8.96 + 8.96i)T - 97iT^{2} \)
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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

−10.08426537854204084959009325151, −9.742649397746404022633148539655, −8.651311180159982403240397438473, −7.76803838802251444570790122539, −6.41058775674964276929861314917, −5.78495949503928300949016402006, −5.23637348232902371503237353860, −4.49508048228725382836111915018, −3.19853846652732830382746052165, −2.04256852080230838845645817627, 0.16818359120258297262607055131, 1.62288934624792898628235204045, 1.97280233893962680630913530838, 4.58554953667355567440933938291, 4.69797792244191729459878791683, 5.76576387563828109826336382698, 6.85894624533566909839350593783, 7.08172390869721145448241168798, 8.088419984679379589967438121519, 8.922447115368796530950684039284

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