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Properties

Label 2-780-13.4-c1-0-1
Degree $2$
Conductor $780$
Sign $-0.999 - 0.00641i$
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  + (−0.5 + 0.866i)3-s + i·5-s + (0.0590 − 0.0340i)7-s + (−0.499 − 0.866i)9-s + (−3.59 − 2.07i)11-s + (−1.86 + 3.08i)13-s + (−0.866 − 0.5i)15-s + (1.27 + 2.21i)17-s + (−6.89 + 3.98i)19-s + 0.0681i·21-s + (2.96 − 5.14i)23-s − 25-s + 0.999·27-s + (−1.80 + 3.12i)29-s + 1.42i·31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + 0.447i·5-s + (0.0223 − 0.0128i)7-s + (−0.166 − 0.288i)9-s + (−1.08 − 0.625i)11-s + (−0.516 + 0.856i)13-s + (−0.223 − 0.129i)15-s + (0.309 + 0.536i)17-s + (−1.58 + 0.913i)19-s + 0.0148i·21-s + (0.619 − 1.07i)23-s − 0.200·25-s + 0.192·27-s + (−0.335 + 0.581i)29-s + 0.256i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(780\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.999 - 0.00641i$
Analytic conductor: \(6.22833\)
Root analytic conductor: \(2.49566\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{780} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 780,\ (\ :1/2),\ -0.999 - 0.00641i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00128703 + 0.401453i\)
\(L(\frac12)\) \(\approx\) \(0.00128703 + 0.401453i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 - iT \)
13 \( 1 + (1.86 - 3.08i)T \)
good7 \( 1 + (-0.0590 + 0.0340i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.59 + 2.07i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.27 - 2.21i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.89 - 3.98i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.96 + 5.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.80 - 3.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.42iT - 31T^{2} \)
37 \( 1 + (9.87 + 5.69i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.807 + 0.465i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.23 + 2.13i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.63iT - 47T^{2} \)
53 \( 1 + 6.36T + 53T^{2} \)
59 \( 1 + (0.0773 - 0.0446i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.41 - 5.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.03 + 2.90i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.41 - 3.12i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 13.8iT - 73T^{2} \)
79 \( 1 - 1.19T + 79T^{2} \)
83 \( 1 - 2.38iT - 83T^{2} \)
89 \( 1 + (-9.04 - 5.22i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.15 + 3.55i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.52643307042258596762414974448, −10.25663913049918670745226525373, −8.952267443014790458689434862264, −8.320401143220317146883419411968, −7.18619772103263667040477417697, −6.30352081725946566025861726787, −5.40218857270864913981640210302, −4.39403568117825330785079330240, −3.35396272808966752071713157188, −2.10176890598852736596873502744, 0.19335769029523642491072894949, 1.94078705551874945981544256667, 3.09443318278187612874927881751, 4.75466034623525292135142028367, 5.23714674537796480068054965075, 6.39474996339170932292578322864, 7.40682587061090262827244618373, 7.989014047196030567705576079246, 8.985122793077705529354188507178, 9.970412771423983902347450258886

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