↑

Properties

Label 2-308-28.19-c1-0-21
Degree $2$
Conductor $308$
Sign $0.693 + 0.720i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.408 − 1.35i)2-s + (0.276 + 0.478i)3-s + (−1.66 − 1.10i)4-s + (2.05 + 1.18i)5-s + (0.760 − 0.178i)6-s + (2.15 + 1.52i)7-s + (−2.17 + 1.80i)8-s + (1.34 − 2.33i)9-s + (2.44 − 2.29i)10-s + (−0.866 + 0.5i)11-s + (0.0683 − 1.10i)12-s + 1.59i·13-s + (2.95 − 2.29i)14-s + 1.30i·15-s + (1.55 + 3.68i)16-s + (3.13 − 1.80i)17-s + ⋯
L(s)  = 1  + (0.288 − 0.957i)2-s + (0.159 + 0.276i)3-s + (−0.833 − 0.552i)4-s + (0.917 + 0.529i)5-s + (0.310 − 0.0729i)6-s + (0.816 + 0.577i)7-s + (−0.769 + 0.638i)8-s + (0.449 − 0.777i)9-s + (0.771 − 0.725i)10-s + (−0.261 + 0.150i)11-s + (0.0197 − 0.318i)12-s + 0.443i·13-s + (0.788 − 0.614i)14-s + 0.337i·15-s + (0.389 + 0.921i)16-s + (0.759 − 0.438i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.693 + 0.720i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.693 + 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.693 + 0.720i$
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: \((2,\ 308,\ (\ :1/2),\ 0.693 + 0.720i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62741 - 0.692775i\)
\(L(\frac12)\) \(\approx\) \(1.62741 - 0.692775i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.408 + 1.35i)T \)
7 \( 1 + (-2.15 - 1.52i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
good3 \( 1 + (-0.276 - 0.478i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.05 - 1.18i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 1.59iT - 13T^{2} \)
17 \( 1 + (-3.13 + 1.80i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.876 + 1.51i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.50 + 2.02i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.35T + 29T^{2} \)
31 \( 1 + (4.05 + 7.02i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.37 - 5.85i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.28iT - 41T^{2} \)
43 \( 1 - 0.306iT - 43T^{2} \)
47 \( 1 + (4.89 - 8.48i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.64 - 4.57i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.38 + 9.32i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.94 - 1.12i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.68 - 4.43i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.85iT - 71T^{2} \)
73 \( 1 + (-4.67 + 2.69i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-13.0 - 7.52i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.27T + 83T^{2} \)
89 \( 1 + (8.88 + 5.12i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.92iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.59210648700264360261212989761, −10.66294264989021755268767489240, −9.684216174133343678686477560846, −9.294953907178930859744623782987, −7.962029882297777694032291416256, −6.37920852624557503730862116888, −5.40283808248562533888685882697, −4.27753285395703997332565775449, −2.89584148040592022982895228675, −1.73140564563899550240179077067, 1.67920298275434439410600809166, 3.77171596634370800726158400155, 5.16094932382162311137780384652, 5.62233833315482853943261318178, 7.14470041992571066623263007112, 7.85264832567917285949042117991, 8.675898448101397502983571843600, 9.857719694966352981042240866895, 10.67988206242076265933125064457, 12.20085059809694801822531670473

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