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Properties

Label 2-308-77.54-c1-0-4
Degree $2$
Conductor $308$
Sign $0.685 + 0.728i$
Analytic cond. $2.45939$
Root an. cond. $1.56824$
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  + (−2.14 + 1.23i)3-s + (−0.349 − 0.201i)5-s + (−0.938 − 2.47i)7-s + (1.56 − 2.71i)9-s + (0.0145 − 3.31i)11-s + 3.56·13-s + 0.999·15-s + (3.51 + 6.09i)17-s + (3.51 − 6.09i)19-s + (5.07 + 4.14i)21-s + (1.29 − 2.24i)23-s + (−2.41 − 4.18i)25-s + 0.341i·27-s − 7.90i·29-s + (−6.85 + 3.95i)31-s + ⋯
L(s)  = 1  + (−1.23 + 0.715i)3-s + (−0.156 − 0.0902i)5-s + (−0.354 − 0.934i)7-s + (0.522 − 0.905i)9-s + (0.00437 − 0.999i)11-s + 0.989·13-s + 0.258·15-s + (0.852 + 1.47i)17-s + (0.806 − 1.39i)19-s + (1.10 + 0.904i)21-s + (0.270 − 0.468i)23-s + (−0.483 − 0.837i)25-s + 0.0657i·27-s − 1.46i·29-s + (−1.23 + 0.710i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.685 + 0.728i$
Analytic conductor: \(2.45939\)
Root analytic conductor: \(1.56824\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (285, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1/2),\ 0.685 + 0.728i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.682210 - 0.294735i\)
\(L(\frac12)\) \(\approx\) \(0.682210 - 0.294735i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (0.938 + 2.47i)T \)
11 \( 1 + (-0.0145 + 3.31i)T \)
good3 \( 1 + (2.14 - 1.23i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.349 + 0.201i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 3.56T + 13T^{2} \)
17 \( 1 + (-3.51 - 6.09i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.51 + 6.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.29 + 2.24i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.90iT - 29T^{2} \)
31 \( 1 + (6.85 - 3.95i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.86 + 4.96i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.56T + 41T^{2} \)
43 \( 1 + 0.937iT - 43T^{2} \)
47 \( 1 + (-7.49 - 4.32i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.992 - 1.71i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.94 - 4.58i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.89 + 5.01i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.426 - 0.738i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.990T + 71T^{2} \)
73 \( 1 + (6.89 + 11.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.95 - 2.28i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.24T + 83T^{2} \)
89 \( 1 + (1.83 + 1.06i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.57iT - 97T^{2} \)
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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

−11.23968521391881920964195549560, −10.80756110893048303962639109812, −10.04999349537897453458304009671, −8.862950053743659760500489827360, −7.67916259165491573516169063488, −6.32578146591673626878301693111, −5.72160724754435150796505081777, −4.40151554996092555135154099917, −3.50999733470198625959197051825, −0.69884957718535691083527582901, 1.50064323044594585941767350321, 3.38499876792192849247414129821, 5.26203981021514092144060654182, 5.74890215635876039417944769848, 6.93849410513162325678088504582, 7.65664097908544381505141576127, 9.125821078390038877345488589009, 10.00041828970656040907559088245, 11.26321551690612734549896680352, 11.85231519961283838674462579074

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