↑

Properties

Label 2-349-349.123-c1-0-10
Degree $2$
Conductor $349$
Sign $-0.889 - 0.457i$
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  + (−2.28 + 1.31i)2-s + (−1.40 + 2.43i)3-s + (2.47 − 4.28i)4-s + (0.108 − 0.188i)5-s − 7.40i·6-s + (3.00 − 1.73i)7-s + 7.77i·8-s + (−2.44 − 4.23i)9-s + 0.573i·10-s + 6.40i·11-s + (6.94 + 12.0i)12-s + (3.09 − 1.78i)13-s + (−4.57 + 7.92i)14-s + (0.305 + 0.529i)15-s + (−5.29 − 9.17i)16-s + 6.82·17-s + ⋯
L(s)  = 1  + (−1.61 + 0.932i)2-s + (−0.810 + 1.40i)3-s + (1.23 − 2.14i)4-s + (0.0486 − 0.0842i)5-s − 3.02i·6-s + (1.13 − 0.655i)7-s + 2.74i·8-s + (−0.814 − 1.41i)9-s + 0.181i·10-s + 1.93i·11-s + (2.00 + 3.47i)12-s + (0.857 − 0.495i)13-s + (−1.22 + 2.11i)14-s + (0.0789 + 0.136i)15-s + (−1.32 − 2.29i)16-s + 1.65·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 - 0.457i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.889 - 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(349\)
Sign: $-0.889 - 0.457i$
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: \((2,\ 349,\ (\ :1/2),\ -0.889 - 0.457i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.126808 + 0.523986i\)
\(L(\frac12)\) \(\approx\) \(0.126808 + 0.523986i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 + (14.6 - 11.5i)T \)
good2 \( 1 + (2.28 - 1.31i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (1.40 - 2.43i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.108 + 0.188i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-3.00 + 1.73i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 6.40iT - 11T^{2} \)
13 \( 1 + (-3.09 + 1.78i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.82T + 17T^{2} \)
19 \( 1 + (2.30 - 3.99i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.65 + 2.87i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.25 + 5.64i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.85T + 31T^{2} \)
37 \( 1 - 0.771T + 37T^{2} \)
41 \( 1 + 4.60T + 41T^{2} \)
43 \( 1 + (-5.76 - 3.32i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.55iT - 47T^{2} \)
53 \( 1 - 3.79iT - 53T^{2} \)
59 \( 1 + (7.78 + 4.49i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 7.60iT - 61T^{2} \)
67 \( 1 + 2.56T + 67T^{2} \)
71 \( 1 + (6.09 - 3.51i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.243 - 0.421i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 3.36iT - 79T^{2} \)
83 \( 1 + (-6.90 - 11.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.16 + 1.24i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.717 - 0.414i)T + (48.5 - 84.0i)T^{2} \)
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

−11.25645470887549115320901233857, −10.39243185556916479115583653919, −10.15769335839053281241138582468, −9.289093497233942515693315929612, −8.092829347239875008972288319722, −7.46106233865563752202006107004, −6.12413929216049050102020656175, −5.22286740557414296956304737583, −4.25080741417305540276475199447, −1.38862154736558458227754771813, 0.826563841054101397838544162712, 1.76169380552073732564199811151, 3.15329858689615213698731535222, 5.57051286153055369668560299956, 6.55629734057863513518345440979, 7.69984178298792945529065300796, 8.380560063804310185943135418075, 8.944219803297822488604873097869, 10.61810570687902387552795095216, 11.10977246773511782854276850664

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