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Properties

Label 1-3751-3751.658-r0-0-0
Degree $1$
Conductor $3751$
Sign $-0.678 - 0.734i$
Analytic cond. $17.4195$
Root an. cond. $17.4195$
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  + (−0.654 + 0.755i)2-s + (−0.104 − 0.994i)3-s + (−0.142 − 0.989i)4-s + (0.988 − 0.151i)5-s + (0.820 + 0.572i)6-s + (0.723 − 0.690i)7-s + (0.841 + 0.540i)8-s + (−0.978 + 0.207i)9-s + (−0.532 + 0.846i)10-s + (−0.969 + 0.244i)12-s + (−0.830 − 0.556i)13-s + (0.0475 + 0.998i)14-s + (−0.254 − 0.967i)15-s + (−0.959 + 0.281i)16-s + (0.00951 − 0.999i)17-s + (0.483 − 0.875i)18-s + ⋯
L(s)  = 1  + (−0.654 + 0.755i)2-s + (−0.104 − 0.994i)3-s + (−0.142 − 0.989i)4-s + (0.988 − 0.151i)5-s + (0.820 + 0.572i)6-s + (0.723 − 0.690i)7-s + (0.841 + 0.540i)8-s + (−0.978 + 0.207i)9-s + (−0.532 + 0.846i)10-s + (−0.969 + 0.244i)12-s + (−0.830 − 0.556i)13-s + (0.0475 + 0.998i)14-s + (−0.254 − 0.967i)15-s + (−0.959 + 0.281i)16-s + (0.00951 − 0.999i)17-s + (0.483 − 0.875i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $-0.678 - 0.734i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (658, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ -0.678 - 0.734i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5341781427 - 1.221676639i\)
\(L(\frac12)\) \(\approx\) \(0.5341781427 - 1.221676639i\)
\(L(1)\) \(\approx\) \(0.8374533808 - 0.3371469112i\)
\(L(1)\) \(\approx\) \(0.8374533808 - 0.3371469112i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.654 + 0.755i)T \)
3 \( 1 + (-0.104 - 0.994i)T \)
5 \( 1 + (0.988 - 0.151i)T \)
7 \( 1 + (0.723 - 0.690i)T \)
13 \( 1 + (-0.830 - 0.556i)T \)
17 \( 1 + (0.00951 - 0.999i)T \)
19 \( 1 + (-0.217 - 0.976i)T \)
23 \( 1 + (-0.564 - 0.825i)T \)
29 \( 1 + (0.198 - 0.980i)T \)
37 \( 1 + (0.345 - 0.938i)T \)
41 \( 1 + (0.123 + 0.992i)T \)
43 \( 1 + (0.449 - 0.893i)T \)
47 \( 1 + (0.974 + 0.226i)T \)
53 \( 1 + (0.997 + 0.0760i)T \)
59 \( 1 + (0.123 - 0.992i)T \)
61 \( 1 + (0.974 + 0.226i)T \)
67 \( 1 + (0.981 + 0.189i)T \)
71 \( 1 + (-0.948 + 0.318i)T \)
73 \( 1 + (0.723 + 0.690i)T \)
79 \( 1 + (0.964 + 0.263i)T \)
83 \( 1 + (0.997 + 0.0760i)T \)
89 \( 1 + (-0.736 - 0.676i)T \)
97 \( 1 + (-0.998 + 0.0570i)T \)
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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

−18.91404256940249968210481316167, −17.94125535170986066721774856048, −17.67377254615226209379623616938, −16.8372834832882242068647196623, −16.51407188199049125505898339269, −15.44713150117294910128188653679, −14.70293543414481866528002821711, −14.18142275216519710442101288684, −13.3457540580189777220770852437, −12.190441998712921539072087390897, −12.03735403953001539456584643912, −10.9037160099621700904454396934, −10.584932500347613863896960690996, −9.74161721809293855352296024452, −9.361839795460894798597561566574, −8.57216215780568087918329196539, −8.01332185358584924231813079699, −6.893082127818545810213168319850, −5.87665118124107039885393747356, −5.278659544302538945839313688, −4.40477363464087939487494453354, −3.68059840079445763601131069235, −2.690053990281858888254027718980, −2.06667209459131515934061954419, −1.33586673807697063098368803496, 0.547250133806343565176925205362, 1.00992681493217813064516538766, 2.34825793189682449507253964905, 2.37607430981738311038819788309, 4.30687668855990602813857432372, 5.14851607490624936057289027888, 5.595706232379285765557990736477, 6.56696694964993294210214638222, 7.03969425075228465480834256735, 7.74593094334796529052173978804, 8.35748849181162652965127821840, 9.170924055127653352609939809895, 9.87802059579921117882949520415, 10.63546800950739462908268816692, 11.27512926380624877437905490313, 12.189870195696324670203883285535, 13.1380023558538257219281318177, 13.6484121673001315669801096835, 14.28824419829747289630678781886, 14.70280431783816288383351217406, 15.7504029366484616472986405939, 16.67902387412209443966748264536, 17.15475101890955054070571360193, 17.72114902170220445586670137320, 18.04208564357648168031012100668

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