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Properties

Label 2-308-28.19-c1-0-25
Degree $2$
Conductor $308$
Sign $-0.497 + 0.867i$
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  + (−1.40 − 0.116i)2-s + (−0.800 − 1.38i)3-s + (1.97 + 0.329i)4-s + (−0.235 − 0.135i)5-s + (0.966 + 2.04i)6-s + (1.80 − 1.93i)7-s + (−2.74 − 0.694i)8-s + (0.217 − 0.376i)9-s + (0.316 + 0.219i)10-s + (−0.866 + 0.5i)11-s + (−1.12 − 2.99i)12-s + 1.72i·13-s + (−2.77 + 2.51i)14-s + 0.435i·15-s + (3.78 + 1.29i)16-s + (3.81 − 2.20i)17-s + ⋯
L(s)  = 1  + (−0.996 − 0.0825i)2-s + (−0.462 − 0.800i)3-s + (0.986 + 0.164i)4-s + (−0.105 − 0.0608i)5-s + (0.394 + 0.836i)6-s + (0.683 − 0.730i)7-s + (−0.969 − 0.245i)8-s + (0.0724 − 0.125i)9-s + (0.0999 + 0.0693i)10-s + (−0.261 + 0.150i)11-s + (−0.324 − 0.865i)12-s + 0.478i·13-s + (−0.741 + 0.671i)14-s + 0.112i·15-s + (0.945 + 0.324i)16-s + (0.926 − 0.534i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.497 + 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.497 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $-0.497 + 0.867i$
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: \((2,\ 308,\ (\ :1/2),\ -0.497 + 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.336165 - 0.580375i\)
\(L(\frac12)\) \(\approx\) \(0.336165 - 0.580375i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.116i)T \)
7 \( 1 + (-1.80 + 1.93i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
good3 \( 1 + (0.800 + 1.38i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.235 + 0.135i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 1.72iT - 13T^{2} \)
17 \( 1 + (-3.81 + 2.20i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.05 + 1.82i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (8.26 + 4.77i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.98T + 29T^{2} \)
31 \( 1 + (0.259 + 0.449i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.87 + 4.97i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.45iT - 41T^{2} \)
43 \( 1 - 4.88iT - 43T^{2} \)
47 \( 1 + (-5.13 + 8.89i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.74 + 3.01i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.17 - 7.22i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-12.4 - 7.18i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.34 - 1.35i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.88iT - 71T^{2} \)
73 \( 1 + (-5.22 + 3.01i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.90 + 1.67i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.3T + 83T^{2} \)
89 \( 1 + (2.25 + 1.29i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.64iT - 97T^{2} \)
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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

−11.54666152182738637238459973384, −10.37419386343976041065017231750, −9.652074238538938647184119985199, −8.308795680773752727470880141575, −7.57478976123151243099065572252, −6.82906667147609546904790909883, −5.75679839281984783214718357762, −4.05453613933991443812768266321, −2.13250144190431899267305270328, −0.72060135649379835147482091864, 1.88123988124412556173273570650, 3.63620417444291735364927363641, 5.35823570934210598367214852169, 5.87916064648113639755492198630, 7.65907494117775019310727182297, 8.097453005731467464536373246643, 9.412741209528125515983322388897, 10.08131831448030294335064592147, 10.93221098356183253333155494515, 11.64610806855955286412382776986

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