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Properties

Label 2-1360-20.3-c1-0-18
Degree $2$
Conductor $1360$
Sign $0.663 - 0.748i$
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  + (−0.250 + 0.250i)3-s + (−1.28 − 1.82i)5-s + (3.34 + 3.34i)7-s + 2.87i·9-s − 5.02i·11-s + (2.21 + 2.21i)13-s + (0.781 + 0.135i)15-s + (−0.707 + 0.707i)17-s + 1.41·19-s − 1.67·21-s + (−2.70 + 2.70i)23-s + (−1.68 + 4.70i)25-s + (−1.47 − 1.47i)27-s + 0.724i·29-s − 2.60i·31-s + ⋯
L(s)  = 1  + (−0.144 + 0.144i)3-s + (−0.575 − 0.817i)5-s + (1.26 + 1.26i)7-s + 0.958i·9-s − 1.51i·11-s + (0.615 + 0.615i)13-s + (0.201 + 0.0350i)15-s + (−0.171 + 0.171i)17-s + 0.324·19-s − 0.366·21-s + (−0.564 + 0.564i)23-s + (−0.337 + 0.941i)25-s + (−0.283 − 0.283i)27-s + 0.134i·29-s − 0.467i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.663 - 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.663 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1360\)    =    \(2^{4} \cdot 5 \cdot 17\)
Sign: $0.663 - 0.748i$
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: \((2,\ 1360,\ (\ :1/2),\ 0.663 - 0.748i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.595065370\)
\(L(\frac12)\) \(\approx\) \(1.595065370\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.28 + 1.82i)T \)
17 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (0.250 - 0.250i)T - 3iT^{2} \)
7 \( 1 + (-3.34 - 3.34i)T + 7iT^{2} \)
11 \( 1 + 5.02iT - 11T^{2} \)
13 \( 1 + (-2.21 - 2.21i)T + 13iT^{2} \)
19 \( 1 - 1.41T + 19T^{2} \)
23 \( 1 + (2.70 - 2.70i)T - 23iT^{2} \)
29 \( 1 - 0.724iT - 29T^{2} \)
31 \( 1 + 2.60iT - 31T^{2} \)
37 \( 1 + (-0.307 + 0.307i)T - 37iT^{2} \)
41 \( 1 - 9.95T + 41T^{2} \)
43 \( 1 + (6.48 - 6.48i)T - 43iT^{2} \)
47 \( 1 + (-2.45 - 2.45i)T + 47iT^{2} \)
53 \( 1 + (-8.16 - 8.16i)T + 53iT^{2} \)
59 \( 1 + 2.80T + 59T^{2} \)
61 \( 1 - 4.65T + 61T^{2} \)
67 \( 1 + (-4.16 - 4.16i)T + 67iT^{2} \)
71 \( 1 + 10.2iT - 71T^{2} \)
73 \( 1 + (-9.59 - 9.59i)T + 73iT^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 + (-7.99 + 7.99i)T - 83iT^{2} \)
89 \( 1 - 12.0iT - 89T^{2} \)
97 \( 1 + (3.69 - 3.69i)T - 97iT^{2} \)
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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

−9.414769826138553027769569786181, −8.742527939032547424807126177534, −8.178520628597587963803755194085, −7.68935446280124338761874385185, −6.06013575007805930979307554542, −5.49198631972684449371593898539, −4.74881999131097527431314057391, −3.81710322964896774187531319884, −2.40995983035232759704225029393, −1.26796314758166878924183649661, 0.77626494668985546383966249194, 2.12851798265056054237026186373, 3.62885319872738728507182757905, 4.19351723934352797404228305795, 5.16134787850147012468069042719, 6.49691414096656204943500619288, 7.11274939992064240582590449365, 7.70865259624640059431208776121, 8.456071031885379360224549999863, 9.699586358771360947796905384624

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