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Properties

Label 1-151-151.39-r0-0-0
Degree $1$
Conductor $151$
Sign $0.478 - 0.877i$
Analytic cond. $0.701241$
Root an. cond. $0.701241$
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  + (0.913 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $0.478 - 0.877i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ 0.478 - 0.877i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.928775880 - 1.144978738i\)
\(L(\frac12)\) \(\approx\) \(1.928775880 - 1.144978738i\)
\(L(1)\) \(\approx\) \(1.853891190 - 0.7630911120i\)
\(L(1)\) \(\approx\) \(1.853891190 - 0.7630911120i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 \)
good2 \( 1 + (0.913 - 0.406i)T \)
3 \( 1 + (0.728 - 0.684i)T \)
5 \( 1 + (0.832 + 0.553i)T \)
7 \( 1 + (-0.999 + 0.0418i)T \)
11 \( 1 + (-0.957 + 0.289i)T \)
13 \( 1 + (-0.570 + 0.821i)T \)
17 \( 1 + (0.463 + 0.886i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.978 - 0.207i)T \)
29 \( 1 + (-0.187 - 0.982i)T \)
31 \( 1 + (0.146 - 0.989i)T \)
37 \( 1 + (0.996 + 0.0836i)T \)
41 \( 1 + (-0.992 + 0.125i)T \)
43 \( 1 + (-0.999 - 0.0418i)T \)
47 \( 1 + (0.604 + 0.796i)T \)
53 \( 1 + (0.876 - 0.481i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (0.228 - 0.973i)T \)
67 \( 1 + (0.0627 + 0.998i)T \)
71 \( 1 + (0.463 - 0.886i)T \)
73 \( 1 + (0.535 + 0.844i)T \)
79 \( 1 + (-0.929 - 0.368i)T \)
83 \( 1 + (0.968 + 0.248i)T \)
89 \( 1 + (-0.268 - 0.963i)T \)
97 \( 1 + (-0.895 - 0.444i)T \)
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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

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

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