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Properties

Label 1-1456-1456.1005-r0-0-0
Degree $1$
Conductor $1456$
Sign $0.335 - 0.942i$
Analytic cond. $6.76163$
Root an. cond. $6.76163$
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  + (−0.866 − 0.5i)3-s + (−0.866 − 0.5i)5-s + (0.5 + 0.866i)9-s + (−0.866 − 0.5i)11-s + (0.5 + 0.866i)15-s + 17-s + (−0.866 + 0.5i)19-s − 23-s + (0.5 + 0.866i)25-s − i·27-s + (0.866 − 0.5i)29-s + (0.5 + 0.866i)31-s + (0.5 + 0.866i)33-s − i·37-s + (−0.5 − 0.866i)41-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s + (−0.866 − 0.5i)5-s + (0.5 + 0.866i)9-s + (−0.866 − 0.5i)11-s + (0.5 + 0.866i)15-s + 17-s + (−0.866 + 0.5i)19-s − 23-s + (0.5 + 0.866i)25-s − i·27-s + (0.866 − 0.5i)29-s + (0.5 + 0.866i)31-s + (0.5 + 0.866i)33-s − i·37-s + (−0.5 − 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.335 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.335 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $0.335 - 0.942i$
Analytic conductor: \(6.76163\)
Root analytic conductor: \(6.76163\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (1005, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1456,\ (0:\ ),\ 0.335 - 0.942i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5651550188 - 0.3986463399i\)
\(L(\frac12)\) \(\approx\) \(0.5651550188 - 0.3986463399i\)
\(L(1)\) \(\approx\) \(0.6293940944 - 0.1625191967i\)
\(L(1)\) \(\approx\) \(0.6293940944 - 0.1625191967i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 - T \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 - iT \)
61 \( 1 + (0.866 - 0.5i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−20.92272390307106112691420425028, −20.1960650243205918029523778000, −19.20759648796062030709886275721, −18.566853139138889110798704123894, −17.799052671906995199461619501953, −17.13699711579661204485346845230, −16.01274136375986881808643739634, −15.84365040055072281059479518806, −14.92132419294983514875694269375, −14.28782499215869743023127288804, −12.95787115628519979982710572850, −12.30772566494561928196351810717, −11.64355317130266274443450840956, −10.78575147640210038872243241079, −10.30251306254609717506629083938, −9.47971488435958677920159820533, −8.25016706783724635483286991642, −7.5622200674683623305638310273, −6.67261970967651773603515093546, −5.87840740232894797283773226416, −4.8833516977952368339591475095, −4.218926266625462952703975875207, −3.33728529424093609542381586438, −2.27799485258302125722153752371, −0.697432108876906758441992420434, 0.48764243711096969535154459725, 1.502678894166532127243746558119, 2.74992240679856800471327928766, 3.89913828812003217908989523521, 4.76120955615491717667441215996, 5.549586199956078967639701683285, 6.31303270638552459554701448684, 7.34667963463155160746928312170, 8.05759404907266696374273869809, 8.56902720655499352383644229946, 10.08872463117466897205976509210, 10.549969871211896745890527743780, 11.57587624445142130578041452802, 12.12295928072015356510374720198, 12.718449222007533503130320545498, 13.54608417764067360693902257563, 14.42521237467150542997769035553, 15.678385723740045273010291608742, 15.9738889374033534330652882847, 16.83051462415301782924097921805, 17.4259533349095176575290507971, 18.47559669237162254274855119470, 18.94409953152537511385847994354, 19.59235151499015395514466755519, 20.62554863008674918419053080212

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