↑

Properties

Label 2-338-13.8-c2-0-7
Degree $2$
Conductor $338$
Sign $0.471 - 0.881i$
Analytic cond. $9.20983$
Root an. cond. $3.03477$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)2-s − 4.77·3-s + 2i·4-s + (−5.88 − 5.88i)5-s + (−4.77 − 4.77i)6-s + (0.251 − 0.251i)7-s + (−2 + 2i)8-s + 13.8·9-s − 11.7i·10-s + (−0.523 + 0.523i)11-s − 9.55i·12-s + 0.503·14-s + (28.1 + 28.1i)15-s − 4·16-s + 2.88i·17-s + (13.8 + 13.8i)18-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s − 1.59·3-s + 0.5i·4-s + (−1.17 − 1.17i)5-s + (−0.795 − 0.795i)6-s + (0.0359 − 0.0359i)7-s + (−0.250 + 0.250i)8-s + 1.53·9-s − 1.17i·10-s + (−0.0476 + 0.0476i)11-s − 0.795i·12-s + 0.0359·14-s + (1.87 + 1.87i)15-s − 0.250·16-s + 0.169i·17-s + (0.767 + 0.767i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $0.471 - 0.881i$
Analytic conductor: \(9.20983\)
Root analytic conductor: \(3.03477\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :1),\ 0.471 - 0.881i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.678341 + 0.406341i\)
\(L(\frac12)\) \(\approx\) \(0.678341 + 0.406341i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
13 \( 1 \)
good3 \( 1 + 4.77T + 9T^{2} \)
5 \( 1 + (5.88 + 5.88i)T + 25iT^{2} \)
7 \( 1 + (-0.251 + 0.251i)T - 49iT^{2} \)
11 \( 1 + (0.523 - 0.523i)T - 121iT^{2} \)
17 \( 1 - 2.88iT - 289T^{2} \)
19 \( 1 + (-7.68 - 7.68i)T + 361iT^{2} \)
23 \( 1 - 22.8iT - 529T^{2} \)
29 \( 1 - 30.4T + 841T^{2} \)
31 \( 1 + (-14.8 - 14.8i)T + 961iT^{2} \)
37 \( 1 + (-29.4 + 29.4i)T - 1.36e3iT^{2} \)
41 \( 1 + (18.2 + 18.2i)T + 1.68e3iT^{2} \)
43 \( 1 - 29.3iT - 1.84e3T^{2} \)
47 \( 1 + (-21.2 + 21.2i)T - 2.20e3iT^{2} \)
53 \( 1 + 85.8T + 2.80e3T^{2} \)
59 \( 1 + (-70.7 + 70.7i)T - 3.48e3iT^{2} \)
61 \( 1 - 15.4T + 3.72e3T^{2} \)
67 \( 1 + (-49.2 - 49.2i)T + 4.48e3iT^{2} \)
71 \( 1 + (-68.9 - 68.9i)T + 5.04e3iT^{2} \)
73 \( 1 + (-50.9 + 50.9i)T - 5.32e3iT^{2} \)
79 \( 1 - 105.T + 6.24e3T^{2} \)
83 \( 1 + (27.2 + 27.2i)T + 6.88e3iT^{2} \)
89 \( 1 + (60.1 - 60.1i)T - 7.92e3iT^{2} \)
97 \( 1 + (-14.3 - 14.3i)T + 9.40e3iT^{2} \)
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

−11.65719258523739278779632840041, −11.02384709176681386237634197267, −9.678402071154080186373304796295, −8.374583954900444553445478748943, −7.55347816065344400472053901711, −6.46819829055602893355638258839, −5.38372702069483263684876555262, −4.76593906228643971752625227808, −3.78895713881710045596132317660, −0.912383952984151588245661206671, 0.55158824496091692063466430918, 2.83925022310800832829408022284, 4.15107006105633668953451623708, 5.06595989407726680792674675561, 6.36247071690712234073246409628, 6.88332800605167646460861699623, 8.115289533114097551595118140193, 9.897575801025462500161046356774, 10.70456552144715453248061054012, 11.27005091738368048003909690897

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