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Properties

Label 2-308-28.19-c1-0-1
Degree $2$
Conductor $308$
Sign $-0.948 - 0.316i$
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  + (−0.477 + 1.33i)2-s + (−1.26 − 2.18i)3-s + (−1.54 − 1.27i)4-s + (0.400 + 0.231i)5-s + (3.51 − 0.637i)6-s + (−2.64 − 0.0795i)7-s + (2.42 − 1.44i)8-s + (−1.68 + 2.92i)9-s + (−0.499 + 0.423i)10-s + (−0.866 + 0.5i)11-s + (−0.828 + 4.98i)12-s + 4.48i·13-s + (1.36 − 3.48i)14-s − 1.16i·15-s + (0.770 + 3.92i)16-s + (−2.82 + 1.63i)17-s + ⋯
L(s)  = 1  + (−0.337 + 0.941i)2-s + (−0.728 − 1.26i)3-s + (−0.772 − 0.635i)4-s + (0.179 + 0.103i)5-s + (1.43 − 0.260i)6-s + (−0.999 − 0.0300i)7-s + (0.858 − 0.512i)8-s + (−0.562 + 0.974i)9-s + (−0.157 + 0.133i)10-s + (−0.261 + 0.150i)11-s + (−0.239 + 1.43i)12-s + 1.24i·13-s + (0.365 − 0.930i)14-s − 0.301i·15-s + (0.192 + 0.981i)16-s + (−0.685 + 0.395i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $-0.948 - 0.316i$
Analytic conductor: \(2.45939\)
Root analytic conductor: \(1.56824\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1/2),\ -0.948 - 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0230295 + 0.141970i\)
\(L(\frac12)\) \(\approx\) \(0.0230295 + 0.141970i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.477 - 1.33i)T \)
7 \( 1 + (2.64 + 0.0795i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
good3 \( 1 + (1.26 + 2.18i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.400 - 0.231i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 4.48iT - 13T^{2} \)
17 \( 1 + (2.82 - 1.63i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.32 - 4.03i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.210 + 0.121i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.47T + 29T^{2} \)
31 \( 1 + (-2.94 - 5.10i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.98 + 6.90i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.1iT - 41T^{2} \)
43 \( 1 + 4.79iT - 43T^{2} \)
47 \( 1 + (-0.0148 + 0.0256i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.40 + 4.17i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.50 + 6.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.56 + 2.05i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.0 - 6.35i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.58iT - 71T^{2} \)
73 \( 1 + (-3.19 + 1.84i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.66 - 4.42i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.91T + 83T^{2} \)
89 \( 1 + (6.98 + 4.03i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.79iT - 97T^{2} \)
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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

−12.38838311250511206132636267348, −11.21032045979559727754566935544, −10.10434679580221774135145280448, −9.172884985236807999841763009090, −8.030882948027958114158496437036, −7.01666708150597650689318880676, −6.41818673900274956924114744680, −5.77504630019513897232570504721, −4.19353292970609059208460503836, −1.83155695659368696726306554961, 0.12342051981712625707480541296, 2.78708739180671781261746259639, 3.90696414567223384407328781667, 4.98841666796900783505432578056, 5.98155670930665609354642088018, 7.62239195576901829335086706100, 9.063731383495282148803927638665, 9.570357551181128901268625292990, 10.46497991569626652172630411369, 10.99101999197377618072389920939

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