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Properties

Label 2-352-44.39-c1-0-9
Degree $2$
Conductor $352$
Sign $0.371 + 0.928i$
Analytic cond. $2.81073$
Root an. cond. $1.67652$
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  + (0.708 − 0.975i)3-s + (0.671 − 2.06i)5-s + (1.18 − 0.858i)7-s + (0.477 + 1.47i)9-s + (−3.29 − 0.371i)11-s + (5.50 − 1.78i)13-s + (−1.54 − 2.11i)15-s + (−2.83 − 0.921i)17-s + (−4.63 − 3.36i)19-s − 1.76i·21-s − 0.400i·23-s + (0.222 + 0.162i)25-s + (5.21 + 1.69i)27-s + (2.26 + 3.11i)29-s + (−1.41 + 0.460i)31-s + ⋯
L(s)  = 1  + (0.409 − 0.563i)3-s + (0.300 − 0.924i)5-s + (0.446 − 0.324i)7-s + (0.159 + 0.490i)9-s + (−0.993 − 0.112i)11-s + (1.52 − 0.496i)13-s + (−0.397 − 0.547i)15-s + (−0.688 − 0.223i)17-s + (−1.06 − 0.773i)19-s − 0.384i·21-s − 0.0834i·23-s + (0.0445 + 0.0324i)25-s + (1.00 + 0.325i)27-s + (0.419 + 0.577i)29-s + (−0.254 + 0.0827i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(352\)    =    \(2^{5} \cdot 11\)
Sign: $0.371 + 0.928i$
Analytic conductor: \(2.81073\)
Root analytic conductor: \(1.67652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{352} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 352,\ (\ :1/2),\ 0.371 + 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34495 - 0.910221i\)
\(L(\frac12)\) \(\approx\) \(1.34495 - 0.910221i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (3.29 + 0.371i)T \)
good3 \( 1 + (-0.708 + 0.975i)T + (-0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.671 + 2.06i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.18 + 0.858i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-5.50 + 1.78i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.83 + 0.921i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.63 + 3.36i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 0.400iT - 23T^{2} \)
29 \( 1 + (-2.26 - 3.11i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.41 - 0.460i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-5.01 + 3.64i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (2.50 - 3.45i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 - 6.17T + 43T^{2} \)
47 \( 1 + (4.79 - 6.59i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.633 + 1.95i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-7.19 - 9.90i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (12.4 + 4.04i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 2.72iT - 67T^{2} \)
71 \( 1 + (-9.62 - 3.12i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.77 - 3.81i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.34 + 7.20i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.31 - 10.1i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 1.16T + 89T^{2} \)
97 \( 1 + (-5.15 - 15.8i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.99091276572895667942904519356, −10.71159209462170840084638606419, −9.161180032827287290618901254768, −8.409261940182769167529954320380, −7.78656902353057562155756131017, −6.54005063987887360482569394160, −5.30142271177874437391274195129, −4.38387248517427633075870587551, −2.63099320147368262214905205406, −1.24114898896623376475622490864, 2.13141761680469463215722684236, 3.42708608729147455232122641181, 4.47292318058643453576010630414, 5.98822870669951759100926882329, 6.70001748084125243998993024282, 8.151769769386247179846771222762, 8.804037303296103468998653629905, 9.956587215132374124696924203812, 10.66199751319597310528108271423, 11.36732863451861794502341115790

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