Properties

Degree 1
Conductor 151
Sign $0.478 + 0.877i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.913 + 0.406i)2-s + (0.728 + 0.684i)3-s + (0.669 + 0.743i)4-s + (0.832 − 0.553i)5-s + (0.387 + 0.921i)6-s + (−0.999 − 0.0418i)7-s + (0.309 + 0.951i)8-s + (0.0627 + 0.998i)9-s + (0.985 − 0.166i)10-s + (−0.957 − 0.289i)11-s + (−0.0209 + 0.999i)12-s + (−0.570 − 0.821i)13-s + (−0.895 − 0.444i)14-s + (0.985 + 0.166i)15-s + (−0.104 + 0.994i)16-s + (0.463 − 0.886i)17-s + ⋯
L(s,χ)  = 1  + (0.913 + 0.406i)2-s + (0.728 + 0.684i)3-s + (0.669 + 0.743i)4-s + (0.832 − 0.553i)5-s + (0.387 + 0.921i)6-s + (−0.999 − 0.0418i)7-s + (0.309 + 0.951i)8-s + (0.0627 + 0.998i)9-s + (0.985 − 0.166i)10-s + (−0.957 − 0.289i)11-s + (−0.0209 + 0.999i)12-s + (−0.570 − 0.821i)13-s + (−0.895 − 0.444i)14-s + (0.985 + 0.166i)15-s + (−0.104 + 0.994i)16-s + (0.463 − 0.886i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.478 + 0.877i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.478 + 0.877i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(151\)
\( \varepsilon \)  =  $0.478 + 0.877i$
motivic weight  =  \(0\)
character  :  $\chi_{151} (31, \cdot )$
Sato-Tate  :  $\mu(75)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 151,\ (0:\ ),\ 0.478 + 0.877i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.928775880 + 1.144978738i$
$L(\frac12,\chi)$  $\approx$  $1.928775880 + 1.144978738i$
$L(\chi,1)$  $\approx$  1.853891190 + 0.7630911120i
$L(1,\chi)$  $\approx$  1.853891190 + 0.7630911120i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−28.56389752329593830360818418041, −26.45482887921772572007787821408, −25.79382850967436255309315719506, −24.94694241843311692591824720837, −23.897825901736616117510658189215, −22.98758644089557818954543053497, −21.93111034628257381528188318803, −21.05114002545688479496879561160, −20.07638538601618060440263932161, −18.95861953676829760642846904852, −18.47844254309329243903360132929, −16.79173212567832260875033161988, −15.36045978496273613624075431385, −14.473317426201417485169580555615, −13.57319687937695538811883439471, −12.86689067976545666247078491391, −11.89527397104021886248530395254, −10.15802403508718373407529090224, −9.66588109675863615939882921047, −7.728751291246156889769323527703, −6.54027314649018135025939008260, −5.7933927578014927058481418014, −3.890771227620854631070112523320, −2.71013400794593067471483673643, −1.8902725744889215843966845781, 2.50955303571923716358118448129, 3.286072370743719282744758564776, 4.902150287134351659722704456736, 5.57918504544601546520192112261, 7.150351931869424812657148598472, 8.38960971941435100006342182193, 9.61929783340504944046994640726, 10.54912820327268652591012295770, 12.33076159735282236127474271383, 13.34732749893590500703934028095, 13.84449741308753619737643091323, 15.177932962344251719012130598994, 16.05086367526025817530952956338, 16.69093378081316359114535625851, 18.13553680160163456578921359801, 19.93699251473531263043181369936, 20.36709348827211278776477492092, 21.64366009955496582601950046411, 22.0051180986343573044876604254, 23.271229828855238840177215587412, 24.508853658590303605707251363543, 25.36069338744487975941335507102, 25.91838766248266519175936355914, 26.90137478608344653167183909661, 28.42148605293766082640819168940

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