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Properties

Label 2-1360-20.7-c1-0-47
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: $\chi_{1360} (1327, \cdot )$
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

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

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