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Properties

Label 1-1339-1339.1033-r0-0-0
Degree $1$
Conductor $1339$
Sign $-0.614 - 0.788i$
Analytic cond. $6.21828$
Root an. cond. $6.21828$
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  + (0.0615 + 0.998i)2-s + (−0.881 − 0.473i)3-s + (−0.992 + 0.122i)4-s + (0.673 + 0.739i)5-s + (0.417 − 0.908i)6-s + (0.759 + 0.650i)7-s + (−0.183 − 0.982i)8-s + (0.552 + 0.833i)9-s + (−0.696 + 0.717i)10-s + (0.0615 + 0.998i)11-s + (0.932 + 0.361i)12-s + (−0.602 + 0.798i)14-s + (−0.243 − 0.969i)15-s + (0.969 − 0.243i)16-s + (−0.816 − 0.577i)17-s + (−0.798 + 0.602i)18-s + ⋯
L(s)  = 1  + (0.0615 + 0.998i)2-s + (−0.881 − 0.473i)3-s + (−0.992 + 0.122i)4-s + (0.673 + 0.739i)5-s + (0.417 − 0.908i)6-s + (0.759 + 0.650i)7-s + (−0.183 − 0.982i)8-s + (0.552 + 0.833i)9-s + (−0.696 + 0.717i)10-s + (0.0615 + 0.998i)11-s + (0.932 + 0.361i)12-s + (−0.602 + 0.798i)14-s + (−0.243 − 0.969i)15-s + (0.969 − 0.243i)16-s + (−0.816 − 0.577i)17-s + (−0.798 + 0.602i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.614 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.614 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1339\)    =    \(13 \cdot 103\)
Sign: $-0.614 - 0.788i$
Analytic conductor: \(6.21828\)
Root analytic conductor: \(6.21828\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1339} (1033, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1339,\ (0:\ ),\ -0.614 - 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2148136555 + 0.4399607360i\)
\(L(\frac12)\) \(\approx\) \(-0.2148136555 + 0.4399607360i\)
\(L(1)\) \(\approx\) \(0.5604014489 + 0.4702756494i\)
\(L(1)\) \(\approx\) \(0.5604014489 + 0.4702756494i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.0615 + 0.998i)T \)
3 \( 1 + (-0.881 - 0.473i)T \)
5 \( 1 + (0.673 + 0.739i)T \)
7 \( 1 + (0.759 + 0.650i)T \)
11 \( 1 + (0.0615 + 0.998i)T \)
17 \( 1 + (-0.816 - 0.577i)T \)
19 \( 1 + (-0.473 - 0.881i)T \)
23 \( 1 + (-0.552 + 0.833i)T \)
29 \( 1 + (-0.952 - 0.303i)T \)
31 \( 1 + (-0.961 - 0.273i)T \)
37 \( 1 + (-0.988 + 0.153i)T \)
41 \( 1 + (-0.976 - 0.213i)T \)
43 \( 1 + (-0.153 + 0.988i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.850 + 0.526i)T \)
59 \( 1 + (0.943 + 0.332i)T \)
61 \( 1 + (0.816 + 0.577i)T \)
67 \( 1 + (-0.759 + 0.650i)T \)
71 \( 1 + (-0.303 - 0.952i)T \)
73 \( 1 + (0.673 - 0.739i)T \)
79 \( 1 + (0.739 - 0.673i)T \)
83 \( 1 + (-0.183 + 0.982i)T \)
89 \( 1 + (0.122 - 0.992i)T \)
97 \( 1 + (-0.417 + 0.908i)T \)
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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

−20.66481878837352765002105390043, −19.96173500891745110373644580523, −18.81582080932517613553483727287, −18.10546912439398082746294289718, −17.36158169559128644542313709980, −16.85993594996639781763132823040, −16.18942374016114438041519902175, −14.84670064000883552174191115709, −14.133863292683900855113476468249, −13.296516544562520068934788127128, −12.605629701842012335789013732890, −11.83893424019676600310916390046, −10.92022418456509218758815409333, −10.57532063881434852350683702508, −9.75876637002300665814712423017, −8.76497145336005696757769799403, −8.26446883011392066662539724963, −6.68934831883400491789083149756, −5.645149131884903169049476995600, −5.176619150224273533205190789763, −4.15809922980002067858544366346, −3.70185429372120590981525091670, −2.03641216680249994451834241043, −1.34118358553658849350281120079, −0.21096307256820072840270477273, 1.63104811577392191337852160721, 2.38231508879028086404002147593, 4.00865863499748268313781112424, 5.05904637707143127008598369935, 5.46453774258552450175246781007, 6.428382304965640020784122089814, 7.069100090881879803478552947171, 7.64687527004525558745956917657, 8.83349407806851075054752958802, 9.61293087667945077159851388099, 10.524363400336714645708400390224, 11.43693703143765488738934637973, 12.16555651904658583556004174296, 13.287950572007148710624795561821, 13.5750274060126669912813196113, 14.83647357450358793446880983527, 15.13135594945130185039174863282, 16.06758401431621925175161334113, 17.12439174057204236864205129971, 17.62408879888135645676968392601, 18.09860455783642507402745306418, 18.60125339402228711688741055119, 19.609239721862269138757143510988, 20.947915597319073056719135861456, 21.83571977901684773684363116994

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