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Properties

Label 2-349-349.123-c1-0-19
Degree $2$
Conductor $349$
Sign $0.148 + 0.988i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.34 + 0.778i)2-s + (1.56 − 2.70i)3-s + (0.212 − 0.368i)4-s + (−1.71 + 2.97i)5-s + 4.87i·6-s + (0.0378 − 0.0218i)7-s − 2.45i·8-s + (−3.39 − 5.87i)9-s − 5.35i·10-s − 5.81i·11-s + (−0.664 − 1.15i)12-s + (0.607 − 0.350i)13-s + (−0.0340 + 0.0589i)14-s + (5.37 + 9.31i)15-s + (2.33 + 4.04i)16-s + 1.42·17-s + ⋯
L(s)  = 1  + (−0.953 + 0.550i)2-s + (0.903 − 1.56i)3-s + (0.106 − 0.184i)4-s + (−0.768 + 1.33i)5-s + 1.98i·6-s + (0.0143 − 0.00825i)7-s − 0.867i·8-s + (−1.13 − 1.95i)9-s − 1.69i·10-s − 1.75i·11-s + (−0.191 − 0.332i)12-s + (0.168 − 0.0973i)13-s + (−0.00909 + 0.0157i)14-s + (1.38 + 2.40i)15-s + (0.583 + 1.01i)16-s + 0.345·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.148 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.148 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(349\)
Sign: $0.148 + 0.988i$
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: \((2,\ 349,\ (\ :1/2),\ 0.148 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.586640 - 0.505292i\)
\(L(\frac12)\) \(\approx\) \(0.586640 - 0.505292i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 + (-17.9 - 5.19i)T \)
good2 \( 1 + (1.34 - 0.778i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1.56 + 2.70i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.71 - 2.97i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.0378 + 0.0218i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 5.81iT - 11T^{2} \)
13 \( 1 + (-0.607 + 0.350i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.42T + 17T^{2} \)
19 \( 1 + (-2.16 + 3.75i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.24 + 5.62i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.358 + 0.620i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.37T + 31T^{2} \)
37 \( 1 - 0.874T + 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 + (3.09 + 1.78i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 7.39iT - 47T^{2} \)
53 \( 1 + 11.3iT - 53T^{2} \)
59 \( 1 + (-1.98 - 1.14i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 10.0iT - 61T^{2} \)
67 \( 1 - 5.82T + 67T^{2} \)
71 \( 1 + (4.26 - 2.46i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.25 + 9.10i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 2.91iT - 79T^{2} \)
83 \( 1 + (-6.89 - 11.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-10.4 - 6.02i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.42 + 5.44i)T + (48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.35655435326564804981208618574, −10.23183569296444275159722630788, −8.869922681151482919241830638188, −8.177649145378365669083226832908, −7.75424282358399816813854508469, −6.68535142857311214597588222001, −6.36034761897012529866446357702, −3.52082358791154781678105302222, −2.82043567547957633122787877434, −0.68305532840210389357907313636, 1.81100795768940178483155175092, 3.54897308438958310389615407288, 4.61721202807474687392466210909, 5.20616783944536476793128073284, 7.75620239547739618310300655556, 8.309581189630370432499087351078, 9.120019234085704858551087331969, 9.932620598808572043389544578539, 10.11735425352738842223488052402, 11.58953497027236237531872500209

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