↑

Properties

Label 2-349-349.348-c1-0-10
Degree $2$
Conductor $349$
Sign $0.999 - 0.0329i$
Analytic cond. $2.78677$
Root an. cond. $1.66936$
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  + 1.19i·2-s − 2.66·3-s + 0.575·4-s − 1.00·5-s − 3.18i·6-s − 3.16i·7-s + 3.07i·8-s + 4.10·9-s − 1.19i·10-s − 0.0928i·11-s − 1.53·12-s − 5.39i·13-s + 3.77·14-s + 2.67·15-s − 2.51·16-s + 1.78·17-s + ⋯
L(s)  = 1  + 0.843i·2-s − 1.53·3-s + 0.287·4-s − 0.449·5-s − 1.29i·6-s − 1.19i·7-s + 1.08i·8-s + 1.36·9-s − 0.379i·10-s − 0.0280i·11-s − 0.443·12-s − 1.49i·13-s + 1.00·14-s + 0.691·15-s − 0.629·16-s + 0.431·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(349\)
Sign: $0.999 - 0.0329i$
Analytic conductor: \(2.78677\)
Root analytic conductor: \(1.66936\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (348, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 349,\ (\ :1/2),\ 0.999 - 0.0329i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.835174 + 0.0137756i\)
\(L(\frac12)\) \(\approx\) \(0.835174 + 0.0137756i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 + (18.6 - 0.616i)T \)
good2 \( 1 - 1.19iT - 2T^{2} \)
3 \( 1 + 2.66T + 3T^{2} \)
5 \( 1 + 1.00T + 5T^{2} \)
7 \( 1 + 3.16iT - 7T^{2} \)
11 \( 1 + 0.0928iT - 11T^{2} \)
13 \( 1 + 5.39iT - 13T^{2} \)
17 \( 1 - 1.78T + 17T^{2} \)
19 \( 1 - 6.77T + 19T^{2} \)
23 \( 1 - 7.82T + 23T^{2} \)
29 \( 1 + 1.50T + 29T^{2} \)
31 \( 1 - 5.64T + 31T^{2} \)
37 \( 1 + 9.77T + 37T^{2} \)
41 \( 1 - 1.12T + 41T^{2} \)
43 \( 1 + 7.91iT - 43T^{2} \)
47 \( 1 + 12.5iT - 47T^{2} \)
53 \( 1 + 6.18iT - 53T^{2} \)
59 \( 1 + 2.13iT - 59T^{2} \)
61 \( 1 + 3.31iT - 61T^{2} \)
67 \( 1 - 5.45T + 67T^{2} \)
71 \( 1 - 6.90iT - 71T^{2} \)
73 \( 1 - 4.92T + 73T^{2} \)
79 \( 1 - 10.2iT - 79T^{2} \)
83 \( 1 + 12.2T + 83T^{2} \)
89 \( 1 + 11.6iT - 89T^{2} \)
97 \( 1 - 14.9iT - 97T^{2} \)
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

−11.46768419639714418250598270368, −10.68307925913259759716584443989, −10.05818049946107446376248999619, −8.244375589331630671297234360633, −7.28841193826689778311710925909, −6.86515602980561502307652813852, −5.52075563713609262913759092075, −5.13393134295048555591322469613, −3.46064238113226440604112665484, −0.799717261312462173598975235054, 1.37072106421313115241999564949, 3.01169572127327643588532323995, 4.53168719106172001102456030199, 5.60310299984347342142410103289, 6.53077467387273964864577264578, 7.41864664372671963430305525649, 9.097101573718614474936542644305, 9.874165778838600133309388162751, 11.08865725880422152254903383312, 11.49234019793975406136324309887

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