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Properties

Label 1-6003-6003.1175-r0-0-0
Degree $1$
Conductor $6003$
Sign $0.932 - 0.362i$
Analytic cond. $27.8778$
Root an. cond. $27.8778$
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  + (0.991 − 0.128i)2-s + (0.966 − 0.255i)4-s + (−0.984 + 0.175i)5-s + (−0.675 − 0.737i)7-s + (0.925 − 0.377i)8-s + (−0.953 + 0.301i)10-s + (0.0543 − 0.998i)11-s + (−0.288 + 0.957i)13-s + (−0.764 − 0.644i)14-s + (0.869 − 0.494i)16-s + (−0.755 + 0.654i)17-s + (0.965 + 0.262i)19-s + (−0.906 + 0.421i)20-s + (−0.0747 − 0.997i)22-s + (0.938 − 0.346i)25-s + (−0.162 + 0.986i)26-s + ⋯
L(s)  = 1  + (0.991 − 0.128i)2-s + (0.966 − 0.255i)4-s + (−0.984 + 0.175i)5-s + (−0.675 − 0.737i)7-s + (0.925 − 0.377i)8-s + (−0.953 + 0.301i)10-s + (0.0543 − 0.998i)11-s + (−0.288 + 0.957i)13-s + (−0.764 − 0.644i)14-s + (0.869 − 0.494i)16-s + (−0.755 + 0.654i)17-s + (0.965 + 0.262i)19-s + (−0.906 + 0.421i)20-s + (−0.0747 − 0.997i)22-s + (0.938 − 0.346i)25-s + (−0.162 + 0.986i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $0.932 - 0.362i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ 0.932 - 0.362i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.574120868 - 0.4826178889i\)
\(L(\frac12)\) \(\approx\) \(2.574120868 - 0.4826178889i\)
\(L(1)\) \(\approx\) \(1.562620774 - 0.2146738763i\)
\(L(1)\) \(\approx\) \(1.562620774 - 0.2146738763i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.991 - 0.128i)T \)
5 \( 1 + (-0.984 + 0.175i)T \)
7 \( 1 + (-0.675 - 0.737i)T \)
11 \( 1 + (0.0543 - 0.998i)T \)
13 \( 1 + (-0.288 + 0.957i)T \)
17 \( 1 + (-0.755 + 0.654i)T \)
19 \( 1 + (0.965 + 0.262i)T \)
31 \( 1 + (0.446 + 0.894i)T \)
37 \( 1 + (0.320 - 0.947i)T \)
41 \( 1 + (0.998 + 0.0475i)T \)
43 \( 1 + (-0.446 + 0.894i)T \)
47 \( 1 + (-0.930 + 0.365i)T \)
53 \( 1 + (0.933 - 0.359i)T \)
59 \( 1 + (-0.928 + 0.371i)T \)
61 \( 1 + (-0.307 - 0.951i)T \)
67 \( 1 + (-0.998 + 0.0543i)T \)
71 \( 1 + (0.970 + 0.242i)T \)
73 \( 1 + (-0.806 + 0.591i)T \)
79 \( 1 + (0.999 - 0.00679i)T \)
83 \( 1 + (0.476 + 0.879i)T \)
89 \( 1 + (0.872 - 0.488i)T \)
97 \( 1 + (-0.798 - 0.601i)T \)
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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

−17.76516833872216041446128957516, −16.830986326692111156607936376686, −16.1791825369943868404274131413, −15.58919728844356054640087582637, −15.14615351377416560151251803465, −14.81507929843800757144605886562, −13.57577073583562983861999127632, −13.21541167991878845732566127863, −12.37492919035821878003150375420, −12.02617530108814728456842886670, −11.49066136042972240078290011188, −10.63051839334556310394541979043, −9.81812866057515293369480892991, −9.09615717162603997551830415945, −8.14943236547003089375596583817, −7.514449869982257218990269324832, −6.98123740140105780619036710294, −6.21496688450670825468970069304, −5.36414824123731178212793856660, −4.79064758314390640571506110647, −4.16679238101291676847253889031, −3.207299890834060022073173764691, −2.81067299812530649103269841196, −1.95883086693950311819854353385, −0.66374705740180285409411967348, 0.69634829778639733516553103597, 1.60413918833770866482697539931, 2.78253882046123146994178674462, 3.29972505395742049745653895310, 4.01204187259582048919025356851, 4.430511247617350338388281032119, 5.357254001691904135727160224043, 6.36035881964209002379220407290, 6.65850006305137141293713967180, 7.47861597187329683511617168420, 8.04735780482475983520369006388, 9.047782703295214832671887686972, 9.84890163319586895382500318936, 10.8096719971912733557900989937, 11.054007314917521000111036725103, 11.82845871438551707544758887907, 12.400863541557468097596248155078, 13.13812037792534883608607053226, 13.76718389691702318145686869702, 14.31864269861501887359714652576, 14.93186022800585738162065777948, 15.82781598389297545188011827581, 16.26637892083993968538325780275, 16.54983839357880899801302346856, 17.56636758535777708250981104386

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