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Properties

Label 2-780-65.8-c1-0-1
Degree $2$
Conductor $780$
Sign $0.348 - 0.937i$
Analytic cond. $6.22833$
Root an. cond. $2.49566$
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  + (−0.707 − 0.707i)3-s + (−1.89 − 1.18i)5-s + 0.757i·7-s + 1.00i·9-s + (−0.270 + 0.270i)11-s + (−3.47 + 0.952i)13-s + (0.497 + 2.18i)15-s + (3.66 + 3.66i)17-s + (−1.74 + 1.74i)19-s + (0.535 − 0.535i)21-s + (0.736 − 0.736i)23-s + (2.16 + 4.50i)25-s + (0.707 − 0.707i)27-s − 6.49i·29-s + (7.47 + 7.47i)31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.846 − 0.532i)5-s + 0.286i·7-s + 0.333i·9-s + (−0.0816 + 0.0816i)11-s + (−0.964 + 0.264i)13-s + (0.128 + 0.562i)15-s + (0.887 + 0.887i)17-s + (−0.400 + 0.400i)19-s + (0.116 − 0.116i)21-s + (0.153 − 0.153i)23-s + (0.433 + 0.901i)25-s + (0.136 − 0.136i)27-s − 1.20i·29-s + (1.34 + 1.34i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.348 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(780\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.348 - 0.937i$
Analytic conductor: \(6.22833\)
Root analytic conductor: \(2.49566\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{780} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 780,\ (\ :1/2),\ 0.348 - 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.600387 + 0.417350i\)
\(L(\frac12)\) \(\approx\) \(0.600387 + 0.417350i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (1.89 + 1.18i)T \)
13 \( 1 + (3.47 - 0.952i)T \)
good7 \( 1 - 0.757iT - 7T^{2} \)
11 \( 1 + (0.270 - 0.270i)T - 11iT^{2} \)
17 \( 1 + (-3.66 - 3.66i)T + 17iT^{2} \)
19 \( 1 + (1.74 - 1.74i)T - 19iT^{2} \)
23 \( 1 + (-0.736 + 0.736i)T - 23iT^{2} \)
29 \( 1 + 6.49iT - 29T^{2} \)
31 \( 1 + (-7.47 - 7.47i)T + 31iT^{2} \)
37 \( 1 - 7.43iT - 37T^{2} \)
41 \( 1 + (-1.24 - 1.24i)T + 41iT^{2} \)
43 \( 1 + (8.28 - 8.28i)T - 43iT^{2} \)
47 \( 1 - 3.33iT - 47T^{2} \)
53 \( 1 + (3.05 + 3.05i)T + 53iT^{2} \)
59 \( 1 + (-10.0 - 10.0i)T + 59iT^{2} \)
61 \( 1 + 5.75T + 61T^{2} \)
67 \( 1 - 6.02T + 67T^{2} \)
71 \( 1 + (2.44 + 2.44i)T + 71iT^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 - 1.12iT - 79T^{2} \)
83 \( 1 - 11.6iT - 83T^{2} \)
89 \( 1 + (7.93 + 7.93i)T + 89iT^{2} \)
97 \( 1 - 0.531T + 97T^{2} \)
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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

−10.42943322486220674475624896516, −9.730914145416466270485523713677, −8.451197474335761993481972674400, −8.025939168664436744482755567085, −7.03936697409102711661864687381, −6.09705584722322991866757763186, −5.03482496831703932188436082873, −4.24559935341112177758862753665, −2.88124669209606228136126514000, −1.32068038123845983005643448992, 0.42226260879609703106211546447, 2.65394564393238110594346952543, 3.68479274833860298642210233382, 4.68724201909404126182534371912, 5.56167342711644829018651744092, 6.85731712128965879508605153774, 7.40433970544568936661054582587, 8.350443809331502091427993117395, 9.469872107694547963895871844862, 10.25629137826830145465818556887

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