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Properties

Label 1-19e2-361.303-r1-0-0
Degree $1$
Conductor $361$
Sign $-0.802 - 0.596i$
Analytic cond. $38.7948$
Root an. cond. $38.7948$
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  + (−0.945 − 0.324i)2-s + (0.401 − 0.915i)3-s + (0.789 + 0.614i)4-s + (0.245 + 0.969i)5-s + (−0.677 + 0.735i)6-s + (0.789 − 0.614i)7-s + (−0.546 − 0.837i)8-s + (−0.677 − 0.735i)9-s + (0.0825 − 0.996i)10-s + (−0.677 + 0.735i)11-s + (0.879 − 0.475i)12-s + (0.401 − 0.915i)13-s + (−0.945 + 0.324i)14-s + (0.986 + 0.164i)15-s + (0.245 + 0.969i)16-s + (0.789 − 0.614i)17-s + ⋯
L(s)  = 1  + (−0.945 − 0.324i)2-s + (0.401 − 0.915i)3-s + (0.789 + 0.614i)4-s + (0.245 + 0.969i)5-s + (−0.677 + 0.735i)6-s + (0.789 − 0.614i)7-s + (−0.546 − 0.837i)8-s + (−0.677 − 0.735i)9-s + (0.0825 − 0.996i)10-s + (−0.677 + 0.735i)11-s + (0.879 − 0.475i)12-s + (0.401 − 0.915i)13-s + (−0.945 + 0.324i)14-s + (0.986 + 0.164i)15-s + (0.245 + 0.969i)16-s + (0.789 − 0.614i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $-0.802 - 0.596i$
Analytic conductor: \(38.7948\)
Root analytic conductor: \(38.7948\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (1:\ ),\ -0.802 - 0.596i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3704260580 - 1.118486586i\)
\(L(\frac12)\) \(\approx\) \(0.3704260580 - 1.118486586i\)
\(L(1)\) \(\approx\) \(0.7455473776 - 0.3921144355i\)
\(L(1)\) \(\approx\) \(0.7455473776 - 0.3921144355i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (-0.945 - 0.324i)T \)
3 \( 1 + (0.401 - 0.915i)T \)
5 \( 1 + (0.245 + 0.969i)T \)
7 \( 1 + (0.789 - 0.614i)T \)
11 \( 1 + (-0.677 + 0.735i)T \)
13 \( 1 + (0.401 - 0.915i)T \)
17 \( 1 + (0.789 - 0.614i)T \)
23 \( 1 + (-0.401 - 0.915i)T \)
29 \( 1 + (-0.789 + 0.614i)T \)
31 \( 1 + (-0.945 + 0.324i)T \)
37 \( 1 + (0.677 - 0.735i)T \)
41 \( 1 + (0.986 + 0.164i)T \)
43 \( 1 + (-0.0825 - 0.996i)T \)
47 \( 1 + (-0.677 + 0.735i)T \)
53 \( 1 + (0.677 - 0.735i)T \)
59 \( 1 + (0.986 + 0.164i)T \)
61 \( 1 + (0.546 - 0.837i)T \)
67 \( 1 + (-0.546 - 0.837i)T \)
71 \( 1 + (-0.546 - 0.837i)T \)
73 \( 1 + (0.789 - 0.614i)T \)
79 \( 1 + (0.0825 + 0.996i)T \)
83 \( 1 + (0.245 - 0.969i)T \)
89 \( 1 + (-0.789 - 0.614i)T \)
97 \( 1 + (-0.546 + 0.837i)T \)
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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

−25.043035677692687454270554871900, −24.153352330231893465041578336441, −23.53808658114988257712932384902, −21.69445432226281346597059491487, −21.16774110136139981393750241335, −20.58922685045003976901715404744, −19.52523035211541490609051095565, −18.68616648278333743967545259619, −17.64312761912387599733544054602, −16.61315854277386077517289896528, −16.21770289759203771031418675843, −15.22060738907743881380710746043, −14.41394588968530289990878788534, −13.33340677611029649524563397919, −11.742164628983637905101647223912, −11.09486079522645489309637633227, −9.90181043148939246768720335553, −9.16593519479422017679225682404, −8.37649375961450202512003110626, −7.805126699747310640216141896, −5.8163076944482299762093824569, −5.38582190844132218297967518254, −4.02881002813166400835082217193, −2.4274678650086091189604175315, −1.353291605077636160247527142223, 0.427291031654747539864704604197, 1.71255987834176025359271686861, 2.60041033410735639877047125634, 3.619119442916133514823897115361, 5.625432206374887121414775922299, 6.91987852026074186641917621107, 7.553476060586742750237299419489, 8.16750013347564496506525851726, 9.501955689309496420565903979925, 10.521561560061849654114170976926, 11.15623584251045822069963940355, 12.3255404608084165369209898028, 13.18674138328293693087130477169, 14.37846583571098481908875040768, 15.02086456777229491397887807604, 16.37129629606138087236528822688, 17.60829610413137988940730367667, 18.10430964142505117721831051835, 18.53961164036270437393466582085, 19.70239229190708146974450207662, 20.487351100263105103996664094891, 21.046871141806862499683461996794, 22.49797498978029155721870356172, 23.37069017852977168972713369286, 24.35627860350651762775577674913

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