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Properties

Label 2-780-65.57-c1-0-0
Degree $2$
Conductor $780$
Sign $-0.756 - 0.653i$
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 + (−1.56 + 1.59i)5-s + 2.82i·7-s − 1.00i·9-s + (−0.572 − 0.572i)11-s + (−3.60 + 0.0122i)13-s + (0.0169 + 2.23i)15-s + (−4.83 + 4.83i)17-s + (−5.69 − 5.69i)19-s + (1.99 + 1.99i)21-s + (4.33 + 4.33i)23-s + (−0.0759 − 4.99i)25-s + (−0.707 − 0.707i)27-s + 0.360i·29-s + (−3.47 + 3.47i)31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.701 + 0.712i)5-s + 1.06i·7-s − 0.333i·9-s + (−0.172 − 0.172i)11-s + (−0.999 + 0.00339i)13-s + (0.00438 + 0.577i)15-s + (−1.17 + 1.17i)17-s + (−1.30 − 1.30i)19-s + (0.436 + 0.436i)21-s + (0.904 + 0.904i)23-s + (−0.0151 − 0.999i)25-s + (−0.136 − 0.136i)27-s + 0.0669i·29-s + (−0.624 + 0.624i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 - 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.756 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(780\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.756 - 0.653i$
Analytic conductor: \(6.22833\)
Root analytic conductor: \(2.49566\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{780} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 780,\ (\ :1/2),\ -0.756 - 0.653i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.228590 + 0.614539i\)
\(L(\frac12)\) \(\approx\) \(0.228590 + 0.614539i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.56 - 1.59i)T \)
13 \( 1 + (3.60 - 0.0122i)T \)
good7 \( 1 - 2.82iT - 7T^{2} \)
11 \( 1 + (0.572 + 0.572i)T + 11iT^{2} \)
17 \( 1 + (4.83 - 4.83i)T - 17iT^{2} \)
19 \( 1 + (5.69 + 5.69i)T + 19iT^{2} \)
23 \( 1 + (-4.33 - 4.33i)T + 23iT^{2} \)
29 \( 1 - 0.360iT - 29T^{2} \)
31 \( 1 + (3.47 - 3.47i)T - 31iT^{2} \)
37 \( 1 + 3.02iT - 37T^{2} \)
41 \( 1 + (6.80 - 6.80i)T - 41iT^{2} \)
43 \( 1 + (0.183 + 0.183i)T + 43iT^{2} \)
47 \( 1 - 3.78iT - 47T^{2} \)
53 \( 1 + (-0.953 + 0.953i)T - 53iT^{2} \)
59 \( 1 + (5.73 - 5.73i)T - 59iT^{2} \)
61 \( 1 - 11.1T + 61T^{2} \)
67 \( 1 - 7.59T + 67T^{2} \)
71 \( 1 + (-1.43 + 1.43i)T - 71iT^{2} \)
73 \( 1 - 10.9T + 73T^{2} \)
79 \( 1 - 12.7iT - 79T^{2} \)
83 \( 1 - 9.99iT - 83T^{2} \)
89 \( 1 + (9.06 - 9.06i)T - 89iT^{2} \)
97 \( 1 + 11.3T + 97T^{2} \)
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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

−10.91752420334884863544828889100, −9.641890748387891899911183937632, −8.731161145022688607050367673521, −8.214330366749735278415316588970, −7.04017040454343121208759386805, −6.57572104248520097915251196228, −5.30262244566909281325331479173, −4.15588446145602350877224351090, −2.90816408134424379648245632454, −2.15427204784220245714448379752, 0.29056385689676966426156675205, 2.20038621667409018408486668686, 3.66673378332928672169465168369, 4.45801762050563578886325366087, 5.10189774284919263165623130745, 6.74679615436421203631092739339, 7.44974712109033916897639396792, 8.319081909344519988025546082414, 9.057588447322244974887592454611, 9.997732019640798972836028754872

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