Properties

Degree 2
Conductor 349
Sign $0.997 - 0.0635i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.788 + 0.455i)2-s + (−1.46 + 2.53i)3-s + (−0.585 + 1.01i)4-s + (1.45 − 2.51i)5-s − 2.66i·6-s + (3.29 − 1.90i)7-s − 2.88i·8-s + (−2.77 − 4.81i)9-s + 2.64i·10-s − 3.72i·11-s + (−1.71 − 2.96i)12-s + (3.22 − 1.86i)13-s + (−1.73 + 3.00i)14-s + (4.24 + 7.35i)15-s + (0.145 + 0.251i)16-s − 5.36·17-s + ⋯
L(s)  = 1  + (−0.557 + 0.322i)2-s + (−0.844 + 1.46i)3-s + (−0.292 + 0.506i)4-s + (0.649 − 1.12i)5-s − 1.08i·6-s + (1.24 − 0.718i)7-s − 1.02i·8-s + (−0.925 − 1.60i)9-s + 0.837i·10-s − 1.12i·11-s + (−0.493 − 0.855i)12-s + (0.895 − 0.516i)13-s + (−0.462 + 0.801i)14-s + (1.09 + 1.90i)15-s + (0.0363 + 0.0629i)16-s − 1.30·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0635i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(349\)
\( \varepsilon \)  =  $0.997 - 0.0635i$
motivic weight  =  \(1\)
character  :  $\chi_{349} (123, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 349,\ (\ :1/2),\ 0.997 - 0.0635i)$
$L(1)$  $\approx$  $0.780421 + 0.0248384i$
$L(\frac12)$  $\approx$  $0.780421 + 0.0248384i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 349$,\(F_p(T)\) is a polynomial of degree 2. If $p = 349$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad349 \( 1 + (-6.70 + 17.4i)T \)
good2 \( 1 + (0.788 - 0.455i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (1.46 - 2.53i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.45 + 2.51i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-3.29 + 1.90i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 3.72iT - 11T^{2} \)
13 \( 1 + (-3.22 + 1.86i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.36T + 17T^{2} \)
19 \( 1 + (-1.62 + 2.80i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.39 + 4.14i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.19 - 3.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.79T + 31T^{2} \)
37 \( 1 + 7.48T + 37T^{2} \)
41 \( 1 + 2.38T + 41T^{2} \)
43 \( 1 + (1.70 + 0.985i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.30iT - 47T^{2} \)
53 \( 1 - 12.2iT - 53T^{2} \)
59 \( 1 + (7.06 + 4.08i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 4.05iT - 61T^{2} \)
67 \( 1 - 15.3T + 67T^{2} \)
71 \( 1 + (-6.81 + 3.93i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.57 - 6.19i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 11.6iT - 79T^{2} \)
83 \( 1 + (-7.11 - 12.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-12.8 - 7.42i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.04 - 2.91i)T + (48.5 - 84.0i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.08834501195487634964136823477, −10.65498418357297539987653857714, −9.538736376199063480822301908575, −8.675228049736655223527421214944, −8.300989504174450801560122849746, −6.55796150005317473519422103280, −5.30706737287084198371373899937, −4.64777368427035053389642128074, −3.72462000870486903852508652125, −0.77411903663868631369222143220, 1.73852016144499491521368936299, 2.07095193559141747375298511399, 4.86556789534492667442155280571, 5.91513034771875075674139724127, 6.56989022317247796461633927835, 7.66540542994759912470651943025, 8.599976494208064883439039464095, 9.816279033439259265542724742352, 10.78526268793177231559877298388, 11.47163218150738747626739280811

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