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Properties

Label 1-26e2-676.271-r0-0-0
Degree $1$
Conductor $676$
Sign $-0.764 + 0.644i$
Analytic cond. $3.13933$
Root an. cond. $3.13933$
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  + (0.845 − 0.534i)3-s + (−0.663 − 0.748i)5-s + (−0.721 + 0.692i)7-s + (0.428 − 0.903i)9-s + (−0.903 + 0.428i)11-s + (−0.960 − 0.278i)15-s + (−0.692 − 0.721i)17-s + (−0.866 + 0.5i)19-s + (−0.239 + 0.970i)21-s + (−0.5 + 0.866i)23-s + (−0.120 + 0.992i)25-s + (−0.120 − 0.992i)27-s + (−0.996 + 0.0804i)29-s + (0.992 − 0.120i)31-s + (−0.534 + 0.845i)33-s + ⋯
L(s)  = 1  + (0.845 − 0.534i)3-s + (−0.663 − 0.748i)5-s + (−0.721 + 0.692i)7-s + (0.428 − 0.903i)9-s + (−0.903 + 0.428i)11-s + (−0.960 − 0.278i)15-s + (−0.692 − 0.721i)17-s + (−0.866 + 0.5i)19-s + (−0.239 + 0.970i)21-s + (−0.5 + 0.866i)23-s + (−0.120 + 0.992i)25-s + (−0.120 − 0.992i)27-s + (−0.996 + 0.0804i)29-s + (0.992 − 0.120i)31-s + (−0.534 + 0.845i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $-0.764 + 0.644i$
Analytic conductor: \(3.13933\)
Root analytic conductor: \(3.13933\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 676,\ (0:\ ),\ -0.764 + 0.644i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01005442545 + 0.02753579244i\)
\(L(\frac12)\) \(\approx\) \(0.01005442545 + 0.02753579244i\)
\(L(1)\) \(\approx\) \(0.7878325023 - 0.1682821897i\)
\(L(1)\) \(\approx\) \(0.7878325023 - 0.1682821897i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.845 - 0.534i)T \)
5 \( 1 + (-0.663 - 0.748i)T \)
7 \( 1 + (-0.721 + 0.692i)T \)
11 \( 1 + (-0.903 + 0.428i)T \)
17 \( 1 + (-0.692 - 0.721i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.996 + 0.0804i)T \)
31 \( 1 + (0.992 - 0.120i)T \)
37 \( 1 + (-0.391 + 0.919i)T \)
41 \( 1 + (-0.534 - 0.845i)T \)
43 \( 1 + (-0.919 + 0.391i)T \)
47 \( 1 + (0.935 - 0.354i)T \)
53 \( 1 + (-0.970 + 0.239i)T \)
59 \( 1 + (-0.316 + 0.948i)T \)
61 \( 1 + (0.278 + 0.960i)T \)
67 \( 1 + (0.160 - 0.987i)T \)
71 \( 1 + (-0.999 + 0.0402i)T \)
73 \( 1 + (-0.822 - 0.568i)T \)
79 \( 1 + (0.354 + 0.935i)T \)
83 \( 1 + (-0.464 - 0.885i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (-0.979 - 0.200i)T \)
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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

−22.246921773684437958616356227793, −21.74828566615896575583025424158, −20.624347422781236426654254604576, −20.00413612104807842330774436360, −19.13464567681922142800043014646, −18.7795260438711230142763749888, −17.46519027787723564041143656875, −16.38604090285873950140786352395, −15.71659017953697438640074692605, −15.08252392980246893826283335293, −14.20385543918117989270500602150, −13.35496173837405139670084763982, −12.66766371004532068026266361199, −11.144429665631173402776005131336, −10.57597359675303462747939376989, −9.924229437951724281199274751621, −8.682179365768817575737784868555, −8.00763464021463041739854825241, −7.082327813020646468263351450392, −6.20175008343993100344116502262, −4.64276444809272149099507095646, −3.87617653160446089878737533916, −3.066000306823247499981126984698, −2.19009190546501939262689647401, −0.01164194974963901742818800207, 1.68568155404266846287023500319, 2.66676927566192035301305414718, 3.639419181027727463484023197031, 4.69306341208632925690853316517, 5.83803393957990928954432562021, 6.96525201392373837835317408990, 7.81149567332986192681999083439, 8.59711742779257633771112582660, 9.29614884440539375740630540868, 10.21019435872422812544525220892, 11.7022028190849714338508936868, 12.31576841752299280679956758177, 13.12180498653512804856511703725, 13.641754881000956263586060089255, 15.139947871353972650501130460113, 15.41331253521259889109203921334, 16.27423175845161162798852679995, 17.41173150908568280403659416847, 18.462584862203670843192774621928, 19.00161862207085611564145902319, 19.8039784623896111886860553141, 20.51462653088714555191504584991, 21.15076280873183432335627791960, 22.30999095833625770589263297392, 23.27684683160844220540053670626

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