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Properties

Label 1-1003-1003.412-r1-0-0
Degree $1$
Conductor $1003$
Sign $-0.615 + 0.788i$
Analytic cond. $107.787$
Root an. cond. $107.787$
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  + 2-s − i·3-s + 4-s − i·5-s − i·6-s + i·7-s + 8-s − 9-s − i·10-s − i·11-s − i·12-s − 13-s + i·14-s − 15-s + 16-s + ⋯
L(s)  = 1  + 2-s − i·3-s + 4-s − i·5-s − i·6-s + i·7-s + 8-s − 9-s − i·10-s − i·11-s − i·12-s − 13-s + i·14-s − 15-s + 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1003\)    =    \(17 \cdot 59\)
Sign: $-0.615 + 0.788i$
Analytic conductor: \(107.787\)
Root analytic conductor: \(107.787\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1003} (412, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1003,\ (1:\ ),\ -0.615 + 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4148418178 - 0.8502104973i\)
\(L(\frac12)\) \(\approx\) \(-0.4148418178 - 0.8502104973i\)
\(L(1)\) \(\approx\) \(1.323786964 - 0.7283457939i\)
\(L(1)\) \(\approx\) \(1.323786964 - 0.7283457939i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
59 \( 1 \)
good2 \( 1 \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 - iT \)
11 \( 1 + T \)
13 \( 1 - iT \)
19 \( 1 + iT \)
23 \( 1 + T \)
29 \( 1 - T \)
31 \( 1 - iT \)
37 \( 1 - iT \)
41 \( 1 - iT \)
43 \( 1 - T \)
47 \( 1 + iT \)
53 \( 1 - T \)
61 \( 1 \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 - iT \)
79 \( 1 + T \)
83 \( 1 - iT \)
89 \( 1 - iT \)
97 \( 1 - iT \)
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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

−22.18659843071947377443981458778, −21.24158060729709250497356388417, −20.657369134349167541221871558, −19.715694351251793455288307662633, −19.355791077674398325120804389966, −17.66290567372798590127237164700, −17.18289642775097859968529319540, −16.25215293114700932009382681074, −15.39360642773816461124436622717, −14.702194776406429047168698802735, −14.3783105950326992176061512918, −13.40738416605192521464869334112, −12.415626592418086119019095063339, −11.458219429607548220292223999436, −10.70748991458535643757906650744, −10.21957556663421338064645624729, −9.394200450380607061222463248842, −7.707657565201551448645050786325, −7.20632793422403169070936476557, −6.26425399386816929514427418868, −5.23741217738202059277769782332, −4.34026401817333486943098576660, −3.79161954125462858020879283721, −2.819062091713868229757268380311, −1.92190482240001963205679897700, 0.117296787812000493518707774413, 1.385674804122430591248537959318, 2.333778529753082873579383041502, 3.05229172564148120244953895910, 4.48607941044643019195395779421, 5.24593082676846650763587591784, 6.06723264462381166849174479787, 6.65891692557810196413265831251, 8.042260724068743373202904554231, 8.37046541844464357143157449691, 9.52288493759619394456137427823, 10.919429165969664364682779469534, 11.76407906381109865986286625438, 12.38764256572699347984789427001, 12.83454934656239670465050445194, 13.66053880040707347571708869336, 14.47662080054493771191985357329, 15.238458841902144875455380095617, 16.2736478381286470590028288695, 16.86564992958390201477987922755, 17.7177320163264863366990764995, 19.006862998829016975963832505763, 19.320918147565562759833499128506, 20.21796424666414659885768458566, 21.100064125318241129398844814114

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