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Properties

Label 2-352-44.35-c1-0-1
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: $\chi_{352} (255, \cdot )$
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

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

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