↑

Properties

Label 2-352-11.5-c1-0-0
Degree $2$
Conductor $352$
Sign $0.734 - 0.678i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.57 − 1.87i)3-s + (−0.0283 + 0.0874i)5-s + (−3.14 + 2.28i)7-s + (2.20 + 6.78i)9-s + (3.27 + 0.500i)11-s + (−0.805 − 2.47i)13-s + (0.236 − 0.171i)15-s + (−0.163 + 0.504i)17-s + (4.06 + 2.95i)19-s + 12.3·21-s + 3.05·23-s + (4.03 + 2.93i)25-s + (4.06 − 12.5i)27-s + (−5.34 + 3.88i)29-s + (1.47 + 4.53i)31-s + ⋯
L(s)  = 1  + (−1.48 − 1.08i)3-s + (−0.0127 + 0.0390i)5-s + (−1.19 + 0.864i)7-s + (0.734 + 2.26i)9-s + (0.988 + 0.150i)11-s + (−0.223 − 0.687i)13-s + (0.0611 − 0.0443i)15-s + (−0.0397 + 0.122i)17-s + (0.933 + 0.678i)19-s + 2.70·21-s + 0.636·23-s + (0.807 + 0.586i)25-s + (0.782 − 2.40i)27-s + (−0.992 + 0.721i)29-s + (0.264 + 0.814i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(352\)    =    \(2^{5} \cdot 11\)
Sign: $0.734 - 0.678i$
Analytic conductor: \(2.81073\)
Root analytic conductor: \(1.67652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{352} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 352,\ (\ :1/2),\ 0.734 - 0.678i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.546442 + 0.213602i\)
\(L(\frac12)\) \(\approx\) \(0.546442 + 0.213602i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.27 - 0.500i)T \)
good3 \( 1 + (2.57 + 1.87i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.0283 - 0.0874i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (3.14 - 2.28i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (0.805 + 2.47i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.163 - 0.504i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-4.06 - 2.95i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 3.05T + 23T^{2} \)
29 \( 1 + (5.34 - 3.88i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.47 - 4.53i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (6.66 - 4.84i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-2.34 - 1.70i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 6.01T + 43T^{2} \)
47 \( 1 + (-3.53 - 2.56i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.35 - 10.3i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (9.06 - 6.58i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-3.53 + 10.8i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 7.33T + 67T^{2} \)
71 \( 1 + (3.46 - 10.6i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.04 + 1.48i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.66 + 8.20i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.473 - 1.45i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 3.93T + 89T^{2} \)
97 \( 1 + (1.48 + 4.56i)T + (-78.4 + 57.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.81793656174701472176845448941, −10.89480100346971708096721189135, −9.895939200688530552355374554353, −8.824134489205778968704570878395, −7.38366569696086540759095217900, −6.69744947840908089935062920780, −5.87000009151102292775151955165, −5.11203594810322296082721176475, −3.16204792688640880919185728070, −1.35017912953730865932074692676, 0.54486139092777758103500142433, 3.53679816720163977811425372688, 4.32268687809879702972925064795, 5.43492117257594740152599476633, 6.55264839449827484631597194248, 7.02882114520857895014226437668, 9.150820314700898722336519146117, 9.642043284189055665152049431985, 10.47470092652683197815420701914, 11.34250007724031321332282767843

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