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Properties

Label 1-1976-1976.1659-r0-0-0
Degree $1$
Conductor $1976$
Sign $-0.379 + 0.925i$
Analytic cond. $9.17650$
Root an. cond. $9.17650$
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  + (0.766 + 0.642i)3-s + (0.342 + 0.939i)5-s + (0.866 − 0.5i)7-s + (0.173 + 0.984i)9-s + (0.866 + 0.5i)11-s + (−0.342 + 0.939i)15-s + (−0.173 + 0.984i)17-s + (0.984 + 0.173i)21-s + (−0.939 − 0.342i)23-s + (−0.766 + 0.642i)25-s + (−0.5 + 0.866i)27-s + (−0.173 − 0.984i)29-s + (−0.866 + 0.5i)31-s + (0.342 + 0.939i)33-s + (0.766 + 0.642i)35-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)3-s + (0.342 + 0.939i)5-s + (0.866 − 0.5i)7-s + (0.173 + 0.984i)9-s + (0.866 + 0.5i)11-s + (−0.342 + 0.939i)15-s + (−0.173 + 0.984i)17-s + (0.984 + 0.173i)21-s + (−0.939 − 0.342i)23-s + (−0.766 + 0.642i)25-s + (−0.5 + 0.866i)27-s + (−0.173 − 0.984i)29-s + (−0.866 + 0.5i)31-s + (0.342 + 0.939i)33-s + (0.766 + 0.642i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1976 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1976 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1976\)    =    \(2^{3} \cdot 13 \cdot 19\)
Sign: $-0.379 + 0.925i$
Analytic conductor: \(9.17650\)
Root analytic conductor: \(9.17650\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1976} (1659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1976,\ (0:\ ),\ -0.379 + 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.413759215 + 2.107340748i\)
\(L(\frac12)\) \(\approx\) \(1.413759215 + 2.107340748i\)
\(L(1)\) \(\approx\) \(1.408102737 + 0.7463308568i\)
\(L(1)\) \(\approx\) \(1.408102737 + 0.7463308568i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 + (0.342 + 0.939i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + (-0.173 + 0.984i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (-0.173 - 0.984i)T \)
31 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.642 + 0.766i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (-0.984 + 0.173i)T \)
53 \( 1 + (0.939 + 0.342i)T \)
59 \( 1 + (-0.984 - 0.173i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (-0.984 + 0.173i)T \)
71 \( 1 + (0.342 + 0.939i)T \)
73 \( 1 + (0.642 - 0.766i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + (-0.642 - 0.766i)T \)
97 \( 1 + (0.984 + 0.173i)T \)
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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

−19.90295733409962871762824638253, −19.093727470684116426319880597636, −18.096984999927597109224209164954, −17.88888987220775567876801797761, −16.86537306518837390167387809295, −16.13806704283377085252123655875, −15.27358788882340582158703322953, −14.3213899943558179081291782284, −14.02149761977464243208576468775, −13.184653839395510109871013754009, −12.353119182785549747601489817504, −11.8434214610162901021164063836, −11.03804420044432314796157167389, −9.66118250646076267240653650841, −9.10208526880583444149415963198, −8.58211327064546621450771759231, −7.82308837680217374873107887616, −7.01902599691933979079576808708, −5.96818108824042160733101184413, −5.30745780846460004217720888563, −4.28941926523823715110158283667, −3.45957866764533685269184550707, −2.23617939198744407407001455616, −1.697469637052975445275642442715, −0.75338148005353513280928468353, 1.60585240993962861283736594512, 2.10591581761685235314584196219, 3.23514417215683315232590669933, 4.01922756672363355113670150238, 4.58128061660335540597488711887, 5.74923385346656817971479873968, 6.64847001445942627020909721483, 7.504045485892585522068120986328, 8.171858605963707418851506565061, 9.00974623990166521666383834774, 9.95001329176070421453262838729, 10.36920274040700616949983952668, 11.156659352863552653794737158128, 11.892371942235079474635931233306, 13.12356706752710568484739560897, 13.84233692789815546417030119573, 14.52491784023603792028924574685, 14.84131886611797357213620441504, 15.57059068767623065522577863216, 16.66211300543001221326788997638, 17.28652719814212193713021496857, 17.99855888754528623186695641584, 18.82227306944558863139153272971, 19.65002823402930839058265794515, 20.15850408838425476208615087661

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