↑

Properties

Label 2-1352-13.3-c1-0-35
Degree $2$
Conductor $1352$
Sign $-0.477 + 0.878i$
Analytic cond. $10.7957$
Root an. cond. $3.28569$
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  + (−0.5 − 0.866i)3-s + 5-s + (2.5 − 4.33i)7-s + (1 − 1.73i)9-s + (−1 − 1.73i)11-s + (−0.5 − 0.866i)15-s + (1.5 − 2.59i)17-s + (−1 + 1.73i)19-s − 5·21-s + (−2 − 3.46i)23-s − 4·25-s − 5·27-s + (3 + 5.19i)29-s + 4·31-s + (−0.999 + 1.73i)33-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + 0.447·5-s + (0.944 − 1.63i)7-s + (0.333 − 0.577i)9-s + (−0.301 − 0.522i)11-s + (−0.129 − 0.223i)15-s + (0.363 − 0.630i)17-s + (−0.229 + 0.397i)19-s − 1.09·21-s + (−0.417 − 0.722i)23-s − 0.800·25-s − 0.962·27-s + (0.557 + 0.964i)29-s + 0.718·31-s + (−0.174 + 0.301i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1352\)    =    \(2^{3} \cdot 13^{2}\)
Sign: $-0.477 + 0.878i$
Analytic conductor: \(10.7957\)
Root analytic conductor: \(3.28569\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1352} (1329, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1352,\ (\ :1/2),\ -0.477 + 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.705765210\)
\(L(\frac12)\) \(\approx\) \(1.705765210\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - T + 5T^{2} \)
7 \( 1 + (-2.5 + 4.33i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-5.5 - 9.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4 - 6.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 9T + 47T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3 - 5.19i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.5 + 6.06i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 - 16T + 83T^{2} \)
89 \( 1 + (5 + 8.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5 - 8.66i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.599525352473124127169844373811, −8.121300936278992292384728203382, −7.87726789146238762715955008572, −6.75363348897079475823065785259, −6.29573227746206999082937061923, −5.03815395643918309518748986800, −4.29108359767455476537091642877, −3.20376293241790886983210112969, −1.62712338676556952675748799845, −0.75371846909473262727840231723, 1.85693700839219984660689538115, 2.44677094516806791210642899994, 4.07814090755880214648945266389, 4.98051144552532696103142242950, 5.57644483995376312148707495966, 6.28171384778976307342254895680, 7.78651675301647908062050144506, 8.108668184087682630282890321462, 9.293951780450625961603546924698, 9.718383272404965103631726780028

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