↑

Properties

Label 2-349-349.123-c1-0-22
Degree $2$
Conductor $349$
Sign $0.402 + 0.915i$
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  + (−1.85 + 1.07i)2-s + (1.18 − 2.05i)3-s + (1.29 − 2.23i)4-s + (1.34 − 2.33i)5-s + 5.08i·6-s + (2.87 − 1.66i)7-s + 1.24i·8-s + (−1.31 − 2.28i)9-s + 5.76i·10-s + 0.545i·11-s + (−3.06 − 5.30i)12-s + (0.131 − 0.0758i)13-s + (−3.55 + 6.16i)14-s + (−3.19 − 5.53i)15-s + (1.24 + 2.16i)16-s − 2.30·17-s + ⋯
L(s)  = 1  + (−1.31 + 0.756i)2-s + (0.685 − 1.18i)3-s + (0.645 − 1.11i)4-s + (0.602 − 1.04i)5-s + 2.07i·6-s + (1.08 − 0.628i)7-s + 0.440i·8-s + (−0.439 − 0.761i)9-s + 1.82i·10-s + 0.164i·11-s + (−0.884 − 1.53i)12-s + (0.0364 − 0.0210i)13-s + (−0.951 + 1.64i)14-s + (−0.825 − 1.42i)15-s + (0.312 + 0.540i)16-s − 0.558·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(349\)
Sign: $0.402 + 0.915i$
Analytic conductor: \(2.78677\)
Root analytic conductor: \(1.66936\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 349,\ (\ :1/2),\ 0.402 + 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.859104 - 0.560846i\)
\(L(\frac12)\) \(\approx\) \(0.859104 - 0.560846i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 + (-18.6 - 0.308i)T \)
good2 \( 1 + (1.85 - 1.07i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1.18 + 2.05i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.34 + 2.33i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.87 + 1.66i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 0.545iT - 11T^{2} \)
13 \( 1 + (-0.131 + 0.0758i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.30T + 17T^{2} \)
19 \( 1 + (2.50 - 4.34i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.56 + 4.44i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.81 - 4.87i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.33T + 31T^{2} \)
37 \( 1 + 3.26T + 37T^{2} \)
41 \( 1 - 10.0T + 41T^{2} \)
43 \( 1 + (-6.94 - 4.00i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.68iT - 47T^{2} \)
53 \( 1 + 7.63iT - 53T^{2} \)
59 \( 1 + (2.78 + 1.60i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 8.03iT - 61T^{2} \)
67 \( 1 - 8.84T + 67T^{2} \)
71 \( 1 + (13.5 - 7.83i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.17 - 12.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 0.708iT - 79T^{2} \)
83 \( 1 + (3.54 + 6.14i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (12.3 + 7.14i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.04 + 1.18i)T + (48.5 - 84.0i)T^{2} \)
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

−11.06311836003570737872632505933, −10.07081163641598691400544195112, −8.986804776243549986949262138833, −8.387529573920279016657180034299, −7.80290530165622467696264742478, −6.96697534230225022995068878736, −5.90050322154084305654559282439, −4.42812133984087717234847863367, −1.95167137671199961241062689323, −1.11885354997858226569314135017, 2.10113113062338430447392499589, 2.86511450681194174301861946260, 4.36175651962706050550987917842, 5.77499709360448689470254154261, 7.36520803580200050208492223661, 8.442254801490948231632443870452, 9.094951688902082590494624705142, 9.767702532805832828318906646237, 10.69169383522730700186384975862, 11.04679096593713119750171402144

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