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Properties

Label 2-349-349.123-c1-0-7
Degree $2$
Conductor $349$
Sign $-0.990 - 0.140i$
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  + (0.831 − 0.480i)2-s + (−1.62 + 2.81i)3-s + (−0.538 + 0.932i)4-s + (−1.69 + 2.94i)5-s + 3.12i·6-s + (3.20 − 1.85i)7-s + 2.95i·8-s + (−3.77 − 6.54i)9-s + 3.26i·10-s − 2.98i·11-s + (−1.74 − 3.03i)12-s + (−3.77 + 2.18i)13-s + (1.77 − 3.07i)14-s + (−5.51 − 9.55i)15-s + (0.342 + 0.593i)16-s + 6.34·17-s + ⋯
L(s)  = 1  + (0.588 − 0.339i)2-s + (−0.937 + 1.62i)3-s + (−0.269 + 0.466i)4-s + (−0.759 + 1.31i)5-s + 1.27i·6-s + (1.21 − 0.699i)7-s + 1.04i·8-s + (−1.25 − 2.18i)9-s + 1.03i·10-s − 0.899i·11-s + (−0.505 − 0.874i)12-s + (−1.04 + 0.605i)13-s + (0.475 − 0.823i)14-s + (−1.42 − 2.46i)15-s + (0.0857 + 0.148i)16-s + 1.53·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.140i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(349\)
Sign: $-0.990 - 0.140i$
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: \((2,\ 349,\ (\ :1/2),\ -0.990 - 0.140i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0655727 + 0.926257i\)
\(L(\frac12)\) \(\approx\) \(0.0655727 + 0.926257i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 + (10.1 - 15.7i)T \)
good2 \( 1 + (-0.831 + 0.480i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (1.62 - 2.81i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.69 - 2.94i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-3.20 + 1.85i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 2.98iT - 11T^{2} \)
13 \( 1 + (3.77 - 2.18i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.34T + 17T^{2} \)
19 \( 1 + (0.995 - 1.72i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.30 - 2.25i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.93 - 6.82i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.57T + 31T^{2} \)
37 \( 1 - 1.15T + 37T^{2} \)
41 \( 1 + 1.68T + 41T^{2} \)
43 \( 1 + (-1.19 - 0.690i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.06iT - 47T^{2} \)
53 \( 1 + 4.74iT - 53T^{2} \)
59 \( 1 + (-6.75 - 3.89i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 11.6iT - 61T^{2} \)
67 \( 1 + 8.07T + 67T^{2} \)
71 \( 1 + (5.22 - 3.01i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.280 + 0.485i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 9.98iT - 79T^{2} \)
83 \( 1 + (-8.50 - 14.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-14.0 - 8.12i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.72 - 0.996i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.56722764714265616837736433904, −11.09899772379963993951875091102, −10.58308595932760208883414310136, −9.474042421078604813662053685390, −8.132885793564431845348243359229, −7.20458998108398482757487971843, −5.60550578532535939334152179940, −4.78311230328317505283161117383, −3.81673079809317445816638404494, −3.25016078772947295621514562344, 0.62261761719856255360194243112, 1.88008649841473876121673683617, 4.64410880775027192875800617792, 5.18924556088976105969451605813, 5.85734469854824539419174366599, 7.37860447423422393456144461454, 7.78578487547683696206405241322, 8.878174466157085265300377973618, 10.31149292781221759366671751806, 11.76448294970114629973717654318

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