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Properties

Label 1-2933-2933.1118-r0-0-0
Degree $1$
Conductor $2933$
Sign $0.231 - 0.972i$
Analytic cond. $13.6207$
Root an. cond. $13.6207$
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  + (−0.904 + 0.426i)2-s + (0.592 + 0.805i)3-s + (0.635 − 0.771i)4-s + (−0.0275 − 0.999i)5-s + (−0.879 − 0.475i)6-s + (−0.245 + 0.969i)8-s + (−0.298 + 0.954i)9-s + (0.451 + 0.892i)10-s + (−0.635 + 0.771i)11-s + (0.998 + 0.0550i)12-s − 13-s + (0.789 − 0.614i)15-s + (−0.191 − 0.981i)16-s + (0.0275 − 0.999i)17-s + (−0.137 − 0.990i)18-s + (0.716 − 0.697i)19-s + ⋯
L(s)  = 1  + (−0.904 + 0.426i)2-s + (0.592 + 0.805i)3-s + (0.635 − 0.771i)4-s + (−0.0275 − 0.999i)5-s + (−0.879 − 0.475i)6-s + (−0.245 + 0.969i)8-s + (−0.298 + 0.954i)9-s + (0.451 + 0.892i)10-s + (−0.635 + 0.771i)11-s + (0.998 + 0.0550i)12-s − 13-s + (0.789 − 0.614i)15-s + (−0.191 − 0.981i)16-s + (0.0275 − 0.999i)17-s + (−0.137 − 0.990i)18-s + (0.716 − 0.697i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2933 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.231 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2933 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.231 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2933\)    =    \(7 \cdot 419\)
Sign: $0.231 - 0.972i$
Analytic conductor: \(13.6207\)
Root analytic conductor: \(13.6207\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2933} (1118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2933,\ (0:\ ),\ 0.231 - 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3730868215 - 0.2947321343i\)
\(L(\frac12)\) \(\approx\) \(0.3730868215 - 0.2947321343i\)
\(L(1)\) \(\approx\) \(0.6660121553 + 0.1545895699i\)
\(L(1)\) \(\approx\) \(0.6660121553 + 0.1545895699i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
419 \( 1 \)
good2 \( 1 + (-0.904 + 0.426i)T \)
3 \( 1 + (0.592 + 0.805i)T \)
5 \( 1 + (-0.0275 - 0.999i)T \)
11 \( 1 + (-0.635 + 0.771i)T \)
13 \( 1 - T \)
17 \( 1 + (0.0275 - 0.999i)T \)
19 \( 1 + (0.716 - 0.697i)T \)
23 \( 1 + (-0.592 + 0.805i)T \)
29 \( 1 + (0.546 + 0.837i)T \)
31 \( 1 + (-0.754 + 0.656i)T \)
37 \( 1 + (0.993 - 0.110i)T \)
41 \( 1 + (-0.245 + 0.969i)T \)
43 \( 1 + (-0.401 - 0.915i)T \)
47 \( 1 + (0.926 + 0.376i)T \)
53 \( 1 + (-0.716 - 0.697i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.716 - 0.697i)T \)
67 \( 1 + (-0.451 - 0.892i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.975 + 0.218i)T \)
79 \( 1 + (0.350 + 0.936i)T \)
83 \( 1 + (0.945 - 0.324i)T \)
89 \( 1 + (-0.592 + 0.805i)T \)
97 \( 1 + (0.986 + 0.164i)T \)
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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

−19.06283021013045631778631277012, −18.72815871887528733015020597344, −18.07092585469676640048066421970, −17.47280848280924344001858138351, −16.657228191914962090674695382313, −15.81530047044513756043598195461, −14.92168694763914514573836695123, −14.44998898550554584342424295344, −13.50561181630180451699586831941, −12.86958863887409366333787727533, −11.966375888827842123679135032784, −11.573087879390212977178713600770, −10.44558843067357992039899551837, −10.15501732284946035215981255801, −9.194337607355257624052572499829, −8.35672344541583516287584927117, −7.685039722782276230166878077596, −7.37654951374891589938684626176, −6.29276759592486303929277977407, −5.87275837178195830662877964198, −4.104871337712342245184365577956, −3.3555288762402737176030561568, −2.550463588630470643562273455503, −2.17463494748290133963811686064, −1.0073882593971443679701948421, 0.19266311318434064120346232300, 1.53605909513286211652428209095, 2.36960281930928209486437757965, 3.183124103258446617115533152362, 4.55046063493205327989279298173, 5.055289826409464090189905496011, 5.54469737437124523148694547326, 7.01531607552940798772200850483, 7.56680767962729230228609658592, 8.22580464501907850051929906866, 9.041995333919051013062175346128, 9.67475363599119104947562311187, 9.88804980562544967412333759766, 10.9500877898338754174006813528, 11.720298054558790006939166979351, 12.53577833028041102779508970793, 13.53529985832574713516708588562, 14.22636434280592258051641944623, 15.00535794799884391389910233948, 15.686591342473323197031984766871, 16.1430738024723714313370562033, 16.69148296214953829563078294469, 17.61874359248036178011173917170, 18.003952550698006481721656581270, 19.1013950297658768284285563738

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