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Properties

Label 1-919-919.234-r0-0-0
Degree $1$
Conductor $919$
Sign $-0.916 + 0.399i$
Analytic cond. $4.26781$
Root an. cond. $4.26781$
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  + (0.0922 + 0.995i)2-s + (−0.273 − 0.961i)3-s + (−0.982 + 0.183i)4-s + (−0.273 − 0.961i)5-s + (0.932 − 0.361i)6-s + (0.932 − 0.361i)7-s + (−0.273 − 0.961i)8-s + (−0.850 + 0.526i)9-s + (0.932 − 0.361i)10-s + (−0.850 − 0.526i)11-s + (0.445 + 0.895i)12-s + (−0.982 − 0.183i)13-s + (0.445 + 0.895i)14-s + (−0.850 + 0.526i)15-s + (0.932 − 0.361i)16-s + (−0.602 − 0.798i)17-s + ⋯
L(s)  = 1  + (0.0922 + 0.995i)2-s + (−0.273 − 0.961i)3-s + (−0.982 + 0.183i)4-s + (−0.273 − 0.961i)5-s + (0.932 − 0.361i)6-s + (0.932 − 0.361i)7-s + (−0.273 − 0.961i)8-s + (−0.850 + 0.526i)9-s + (0.932 − 0.361i)10-s + (−0.850 − 0.526i)11-s + (0.445 + 0.895i)12-s + (−0.982 − 0.183i)13-s + (0.445 + 0.895i)14-s + (−0.850 + 0.526i)15-s + (0.932 − 0.361i)16-s + (−0.602 − 0.798i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 919 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 919 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(919\)
Sign: $-0.916 + 0.399i$
Analytic conductor: \(4.26781\)
Root analytic conductor: \(4.26781\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{919} (234, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 919,\ (0:\ ),\ -0.916 + 0.399i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01292180582 - 0.06207059543i\)
\(L(\frac12)\) \(\approx\) \(0.01292180582 - 0.06207059543i\)
\(L(1)\) \(\approx\) \(0.6584868236 - 0.05068222873i\)
\(L(1)\) \(\approx\) \(0.6584868236 - 0.05068222873i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad919 \( 1 \)
good2 \( 1 + (0.0922 + 0.995i)T \)
3 \( 1 + (-0.273 - 0.961i)T \)
5 \( 1 + (-0.273 - 0.961i)T \)
7 \( 1 + (0.932 - 0.361i)T \)
11 \( 1 + (-0.850 - 0.526i)T \)
13 \( 1 + (-0.982 - 0.183i)T \)
17 \( 1 + (-0.602 - 0.798i)T \)
19 \( 1 + (-0.273 + 0.961i)T \)
23 \( 1 + (-0.850 - 0.526i)T \)
29 \( 1 + (0.0922 + 0.995i)T \)
31 \( 1 + (0.932 - 0.361i)T \)
37 \( 1 + (-0.982 - 0.183i)T \)
41 \( 1 + (0.445 - 0.895i)T \)
43 \( 1 + (-0.273 + 0.961i)T \)
47 \( 1 + (-0.982 + 0.183i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.982 + 0.183i)T \)
61 \( 1 + (0.932 + 0.361i)T \)
67 \( 1 + (0.739 + 0.673i)T \)
71 \( 1 + (0.932 - 0.361i)T \)
73 \( 1 + (0.739 + 0.673i)T \)
79 \( 1 + (0.932 + 0.361i)T \)
83 \( 1 + (-0.602 + 0.798i)T \)
89 \( 1 + (-0.273 + 0.961i)T \)
97 \( 1 + (-0.850 + 0.526i)T \)
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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

−22.02108260202307264484558248325, −21.57555954131907339691510697700, −21.0057247389658964167861855691, −19.95966775632089149880385199124, −19.42671908254890542377525376149, −18.3398375571100194557486683018, −17.63425687834829653091128157034, −17.22147334541076800418206964836, −15.488155835208854649014519777559, −15.24110736221525231333329265419, −14.40543996392501677115125851200, −13.6068239981154200285604197283, −12.313208340949790106287473438895, −11.63893794335715774279063137965, −11.00757111083528614606038813043, −10.27390284593630553412917344346, −9.71590403133324972955965548831, −8.56845377241926189765917929469, −7.80481887403216618218824700626, −6.41126890650706748510250980970, −5.22156209550928642938930523061, −4.63662849943826652028292259762, −3.79295445975056451503085389518, −2.64212490884587280920974105926, −2.080992463243334911607629854725, 0.0302844069754610239946797682, 1.13158708289164064066783984433, 2.45023933707347240386043145022, 4.034667612249315194194414878, 5.08478053473681088105104779436, 5.38298754226376703921771797690, 6.60768871206124054434209431522, 7.52438811184800484175373002249, 8.13351736872407042577786473523, 8.586374760302320454566093886645, 9.88149473896190928966852008524, 11.030222004900361975614901985451, 12.15090492560503041839213104670, 12.58970469733287960413828011518, 13.60957470977257176421836997956, 14.0682412028746733586080170692, 15.03795274947470174648789446050, 16.14345638702901208174617099676, 16.63425352902986266300370445797, 17.50627636726526926562843653110, 17.99770031792891448637342182939, 18.84133690835349860721350006406, 19.7579964882032726785851581208, 20.64597550732830318123038579503, 21.50485405971293993024491526109

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