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Properties

Label 1-19e2-361.118-r0-0-0
Degree $1$
Conductor $361$
Sign $0.977 - 0.209i$
Analytic cond. $1.67647$
Root an. cond. $1.67647$
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  + (−0.979 − 0.200i)2-s + (0.983 − 0.182i)3-s + (0.919 + 0.393i)4-s + (−0.971 + 0.236i)5-s + (−0.999 − 0.0183i)6-s + (0.451 − 0.892i)7-s + (−0.821 − 0.569i)8-s + (0.933 − 0.359i)9-s + (0.999 − 0.0367i)10-s + (−0.191 + 0.981i)11-s + (0.975 + 0.218i)12-s + (0.870 + 0.492i)13-s + (−0.621 + 0.783i)14-s + (−0.912 + 0.410i)15-s + (0.690 + 0.723i)16-s + (−0.119 + 0.992i)17-s + ⋯
L(s)  = 1  + (−0.979 − 0.200i)2-s + (0.983 − 0.182i)3-s + (0.919 + 0.393i)4-s + (−0.971 + 0.236i)5-s + (−0.999 − 0.0183i)6-s + (0.451 − 0.892i)7-s + (−0.821 − 0.569i)8-s + (0.933 − 0.359i)9-s + (0.999 − 0.0367i)10-s + (−0.191 + 0.981i)11-s + (0.975 + 0.218i)12-s + (0.870 + 0.492i)13-s + (−0.621 + 0.783i)14-s + (−0.912 + 0.410i)15-s + (0.690 + 0.723i)16-s + (−0.119 + 0.992i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $0.977 - 0.209i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ 0.977 - 0.209i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.098695026 - 0.1162294309i\)
\(L(\frac12)\) \(\approx\) \(1.098695026 - 0.1162294309i\)
\(L(1)\) \(\approx\) \(0.9290732509 - 0.09262388544i\)
\(L(1)\) \(\approx\) \(0.9290732509 - 0.09262388544i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (-0.979 - 0.200i)T \)
3 \( 1 + (0.983 - 0.182i)T \)
5 \( 1 + (-0.971 + 0.236i)T \)
7 \( 1 + (0.451 - 0.892i)T \)
11 \( 1 + (-0.191 + 0.981i)T \)
13 \( 1 + (0.870 + 0.492i)T \)
17 \( 1 + (-0.119 + 0.992i)T \)
23 \( 1 + (-0.333 - 0.942i)T \)
29 \( 1 + (0.957 + 0.289i)T \)
31 \( 1 + (0.0275 + 0.999i)T \)
37 \( 1 + (0.945 - 0.324i)T \)
41 \( 1 + (-0.562 - 0.826i)T \)
43 \( 1 + (-0.467 - 0.883i)T \)
47 \( 1 + (0.515 - 0.856i)T \)
53 \( 1 + (0.484 + 0.875i)T \)
59 \( 1 + (-0.435 + 0.900i)T \)
61 \( 1 + (0.967 - 0.254i)T \)
67 \( 1 + (0.577 + 0.816i)T \)
71 \( 1 + (0.967 + 0.254i)T \)
73 \( 1 + (-0.800 - 0.599i)T \)
79 \( 1 + (-0.531 + 0.847i)T \)
83 \( 1 + (0.993 - 0.110i)T \)
89 \( 1 + (-0.800 + 0.599i)T \)
97 \( 1 + (0.577 - 0.816i)T \)
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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

−24.89385096354053083102332259702, −24.250144670476067219133972309489, −23.348835241800941656089878107263, −21.801406234131911988103389886786, −20.89805272868518981774353086625, −20.228739183972590373607940712274, −19.35057955437415992866453233306, −18.65548276488837385944431840887, −18.017003235767487497545934040238, −16.46642216916586323077662478816, −15.73895409275766973731518789091, −15.321603488663182800505735824558, −14.25711176158925081379327473439, −13.07285334747623625202481202195, −11.67252048137570297721592510061, −11.16490411768746244523054640088, −9.80566085569117071210352761636, −8.87547138092037393654428671039, −8.176307621352664661188971585137, −7.715432179159070079371562311668, −6.250586434857923670720542424820, −4.971703131894484428963781899664, −3.43906252524892491596278261382, −2.57145369224820486057199809247, −1.08436026336760442372506514144, 1.16549282492013670976022813405, 2.28896396502569237457713357098, 3.625172712687713885310593889328, 4.30865833377503928823295584994, 6.69917843520013997926259917607, 7.252605556168720321478821264071, 8.24804071118835529453150954943, 8.72058804532561823723613006228, 10.189676182654970599051772064010, 10.69569407888201651752499279967, 11.94877801383114714686751513419, 12.77851432461170450468667777494, 14.081318951288683105309395179102, 15.01886769240131667271765025991, 15.74716473522821191883132823160, 16.71919795550473433710597987081, 17.90204145648132117609316915638, 18.579496288154995938750731687212, 19.5227201657162672465693803151, 20.14013407413708134041199264070, 20.68453145850084064248151028561, 21.71045671314708771975068671626, 23.37706098758561887880993179945, 23.81460905467684858259647805749, 24.942205592686537271032810503503

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