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Properties

Label 2-349-349.123-c1-0-20
Degree $2$
Conductor $349$
Sign $0.436 + 0.899i$
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  + (0.544 − 0.314i)2-s + (1.24 − 2.15i)3-s + (−0.802 + 1.38i)4-s + (−0.0260 + 0.0450i)5-s − 1.56i·6-s + (3.86 − 2.23i)7-s + 2.26i·8-s + (−1.58 − 2.74i)9-s + 0.0327i·10-s − 0.838i·11-s + (1.99 + 3.44i)12-s + (−0.282 + 0.163i)13-s + (1.40 − 2.43i)14-s + (0.0645 + 0.111i)15-s + (−0.891 − 1.54i)16-s − 4.16·17-s + ⋯
L(s)  = 1  + (0.385 − 0.222i)2-s + (0.716 − 1.24i)3-s + (−0.401 + 0.694i)4-s + (−0.0116 + 0.0201i)5-s − 0.637i·6-s + (1.46 − 0.843i)7-s + 0.801i·8-s + (−0.527 − 0.913i)9-s + 0.0103i·10-s − 0.252i·11-s + (0.574 + 0.995i)12-s + (−0.0783 + 0.0452i)13-s + (0.375 − 0.649i)14-s + (0.0166 + 0.0288i)15-s + (−0.222 − 0.385i)16-s − 1.01·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.436 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.436 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(349\)
Sign: $0.436 + 0.899i$
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: \((2,\ 349,\ (\ :1/2),\ 0.436 + 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71080 - 1.07153i\)
\(L(\frac12)\) \(\approx\) \(1.71080 - 1.07153i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 + (-18.6 + 0.394i)T \)
good2 \( 1 + (-0.544 + 0.314i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1.24 + 2.15i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.0260 - 0.0450i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-3.86 + 2.23i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 0.838iT - 11T^{2} \)
13 \( 1 + (0.282 - 0.163i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.16T + 17T^{2} \)
19 \( 1 + (-1.06 + 1.85i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.05 - 1.82i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.848 + 1.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.88T + 31T^{2} \)
37 \( 1 + 6.18T + 37T^{2} \)
41 \( 1 + 9.18T + 41T^{2} \)
43 \( 1 + (-2.02 - 1.16i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 7.16iT - 47T^{2} \)
53 \( 1 - 14.2iT - 53T^{2} \)
59 \( 1 + (-9.70 - 5.60i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 6.56iT - 61T^{2} \)
67 \( 1 - 1.28T + 67T^{2} \)
71 \( 1 + (3.10 - 1.79i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.64 + 9.78i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 9.05iT - 79T^{2} \)
83 \( 1 + (1.01 + 1.76i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.30 + 1.32i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (12.9 - 7.48i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.50490709282484403664749432800, −10.79153856499825481387243827293, −9.019658266431830738598345732526, −8.369785712427689774851786597581, −7.55497220531836438843507886765, −6.98210501123940787442961455203, −5.16320684138096936232669345737, −4.13949781810005445977164753481, −2.79288882274878174599112232488, −1.50072645619144942672316062685, 2.08105036852020923888837890831, 3.77684547995723168034762402214, 4.85952314021938767921489756923, 5.20519316149107162260077674831, 6.72510510286968830451321347817, 8.435071629766888643902812192886, 8.743015578588242373486642505557, 9.827078530354737010460019252537, 10.54091621812807893100573291739, 11.47607594873524539992373635861

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