↑

Properties

Label 2-349-349.348-c1-0-26
Degree $2$
Conductor $349$
Sign $-0.601 + 0.799i$
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  − 1.36i·2-s + 2.13·3-s + 0.134·4-s − 3.98·5-s − 2.92i·6-s − 4.87i·7-s − 2.91i·8-s + 1.57·9-s + 5.44i·10-s + 0.287i·11-s + 0.287·12-s + 3.64i·13-s − 6.66·14-s − 8.52·15-s − 3.71·16-s + 3.67·17-s + ⋯
L(s)  = 1  − 0.965i·2-s + 1.23·3-s + 0.0672·4-s − 1.78·5-s − 1.19i·6-s − 1.84i·7-s − 1.03i·8-s + 0.525·9-s + 1.72i·10-s + 0.0867i·11-s + 0.0830·12-s + 1.01i·13-s − 1.78·14-s − 2.20·15-s − 0.928·16-s + 0.892·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.601 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.601 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(349\)
Sign: $-0.601 + 0.799i$
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: \((2,\ 349,\ (\ :1/2),\ -0.601 + 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.718441 - 1.43957i\)
\(L(\frac12)\) \(\approx\) \(0.718441 - 1.43957i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 + (-11.2 + 14.9i)T \)
good2 \( 1 + 1.36iT - 2T^{2} \)
3 \( 1 - 2.13T + 3T^{2} \)
5 \( 1 + 3.98T + 5T^{2} \)
7 \( 1 + 4.87iT - 7T^{2} \)
11 \( 1 - 0.287iT - 11T^{2} \)
13 \( 1 - 3.64iT - 13T^{2} \)
17 \( 1 - 3.67T + 17T^{2} \)
19 \( 1 - 4.88T + 19T^{2} \)
23 \( 1 + 3.19T + 23T^{2} \)
29 \( 1 + 0.217T + 29T^{2} \)
31 \( 1 - 6.46T + 31T^{2} \)
37 \( 1 - 1.46T + 37T^{2} \)
41 \( 1 - 7.31T + 41T^{2} \)
43 \( 1 - 2.13iT - 43T^{2} \)
47 \( 1 + 2.45iT - 47T^{2} \)
53 \( 1 + 7.01iT - 53T^{2} \)
59 \( 1 - 13.0iT - 59T^{2} \)
61 \( 1 - 5.02iT - 61T^{2} \)
67 \( 1 - 0.963T + 67T^{2} \)
71 \( 1 - 8.75iT - 71T^{2} \)
73 \( 1 - 13.8T + 73T^{2} \)
79 \( 1 - 11.1iT - 79T^{2} \)
83 \( 1 + 10.7T + 83T^{2} \)
89 \( 1 + 0.115iT - 89T^{2} \)
97 \( 1 + 13.4iT - 97T^{2} \)
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

−11.36303541834090923614513290123, −10.30417337902595443370103674353, −9.539346509703402467066656303419, −8.189156229330480510434129916149, −7.52689279849719725530801432227, −6.94457804054889544311794861673, −4.22694600730438528190403736010, −3.84256781138595246459731146152, −2.94469512685499839971945192766, −1.05275438497634490936254390698, 2.64321946791109421708517085118, 3.38462416296918463707770847418, 5.06801889454755154209097758906, 6.07069866049778444729707640776, 7.59366230293911020979773003348, 7.991733177859398515079933760366, 8.511897918169417823392869663910, 9.496158593769468024196846485406, 11.14854499547992870520630834122, 11.96406803376018384674142832596

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