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Properties

Label 1-1441-1441.30-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.844 + 0.535i$
Analytic cond. $6.69197$
Root an. cond. $6.69197$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.215 + 0.976i)2-s + (−0.354 + 0.935i)3-s + (−0.906 + 0.421i)4-s + (−0.354 + 0.935i)5-s + (−0.989 − 0.144i)6-s + (0.989 − 0.144i)7-s + (−0.607 − 0.794i)8-s + (−0.748 − 0.663i)9-s + (−0.989 − 0.144i)10-s + (−0.0724 − 0.997i)12-s + (0.168 − 0.985i)13-s + (0.354 + 0.935i)14-s + (−0.748 − 0.663i)15-s + (0.644 − 0.764i)16-s + (0.926 + 0.377i)17-s + (0.485 − 0.873i)18-s + ⋯
L(s)  = 1  + (0.215 + 0.976i)2-s + (−0.354 + 0.935i)3-s + (−0.906 + 0.421i)4-s + (−0.354 + 0.935i)5-s + (−0.989 − 0.144i)6-s + (0.989 − 0.144i)7-s + (−0.607 − 0.794i)8-s + (−0.748 − 0.663i)9-s + (−0.989 − 0.144i)10-s + (−0.0724 − 0.997i)12-s + (0.168 − 0.985i)13-s + (0.354 + 0.935i)14-s + (−0.748 − 0.663i)15-s + (0.644 − 0.764i)16-s + (0.926 + 0.377i)17-s + (0.485 − 0.873i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.844 + 0.535i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.844 + 0.535i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4193652821 + 1.444584349i\)
\(L(\frac12)\) \(\approx\) \(0.4193652821 + 1.444584349i\)
\(L(1)\) \(\approx\) \(0.6783792475 + 0.8275104097i\)
\(L(1)\) \(\approx\) \(0.6783792475 + 0.8275104097i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.215 + 0.976i)T \)
3 \( 1 + (-0.354 + 0.935i)T \)
5 \( 1 + (-0.354 + 0.935i)T \)
7 \( 1 + (0.989 - 0.144i)T \)
13 \( 1 + (0.168 - 0.985i)T \)
17 \( 1 + (0.926 + 0.377i)T \)
19 \( 1 + (0.926 - 0.377i)T \)
23 \( 1 + (0.943 + 0.331i)T \)
29 \( 1 + (0.215 - 0.976i)T \)
31 \( 1 + (-0.568 + 0.822i)T \)
37 \( 1 + (0.681 + 0.732i)T \)
41 \( 1 + (0.0724 + 0.997i)T \)
43 \( 1 + (0.998 + 0.0483i)T \)
47 \( 1 + (-0.995 - 0.0965i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.527 - 0.849i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (0.998 - 0.0483i)T \)
71 \( 1 + (-0.958 - 0.285i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.779 + 0.626i)T \)
83 \( 1 + (0.120 - 0.992i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.885 - 0.464i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.540987501610246644870776756850, −19.67403922701503304289584054402, −18.87950525486121253780757242038, −18.41485301047142549647077186038, −17.58999949986505492697659339370, −16.84697584584659141839963279049, −16.111538477029881473649159641024, −14.673031564412860944211269210972, −14.15812005464925986806456361055, −13.39042204603321635873821739027, −12.500161974791155957903414886509, −12.08448904101051870626265489219, −11.368652015556386740119931334115, −10.84625814299635314313690521086, −9.44790931467142478829319337597, −8.85944519091060969732754004102, −7.97802191323087282593300154749, −7.31025279175340387112842218933, −5.87300835686902758227142020889, −5.20695762710496327862139977062, −4.56741983486749556355256070212, −3.48619107480917601097922514404, −2.277464958828779166087407300771, −1.43423304304439705173659262732, −0.83911671992757248076286705280, 0.91604408219873575140545770047, 2.968215175361351471361318413947, 3.503239894219380413243520713194, 4.51186090644568048789843569724, 5.24094473836225654350592955652, 5.919725339388623718187783843072, 6.89946170155449702094167057387, 7.83263519606255719839602124909, 8.28555657789680365527279547698, 9.50367515769286760201140961651, 10.16030657780309569939042309800, 11.08819903016134214187115663039, 11.64668693325654048498065284091, 12.70859503466241157387556027029, 13.83256092403676260898149705152, 14.54274112144893965743397818041, 15.00895782832103729724385306176, 15.61686344332621709953300987387, 16.36071701872796332156631848027, 17.302718547893422115656012523040, 17.773620298080935440317690736114, 18.423231742016501327536116374780, 19.46638551322489829853441785860, 20.54353912700921578623493876422, 21.2675811070145135653157532290

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