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Properties

Label 1-367-367.49-r0-0-0
Degree $1$
Conductor $367$
Sign $0.974 - 0.223i$
Analytic cond. $1.70434$
Root an. cond. $1.70434$
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  + (0.600 + 0.799i)2-s + (0.870 − 0.492i)3-s + (−0.279 + 0.960i)4-s + (−0.998 − 0.0514i)5-s + (0.916 + 0.400i)6-s + (0.229 − 0.973i)7-s + (−0.935 + 0.352i)8-s + (0.514 − 0.857i)9-s + (−0.558 − 0.829i)10-s + (−0.376 − 0.926i)11-s + (0.229 + 0.973i)12-s + (0.514 − 0.857i)13-s + (0.916 − 0.400i)14-s + (−0.894 + 0.447i)15-s + (−0.843 − 0.536i)16-s + (0.952 − 0.304i)17-s + ⋯
L(s)  = 1  + (0.600 + 0.799i)2-s + (0.870 − 0.492i)3-s + (−0.279 + 0.960i)4-s + (−0.998 − 0.0514i)5-s + (0.916 + 0.400i)6-s + (0.229 − 0.973i)7-s + (−0.935 + 0.352i)8-s + (0.514 − 0.857i)9-s + (−0.558 − 0.829i)10-s + (−0.376 − 0.926i)11-s + (0.229 + 0.973i)12-s + (0.514 − 0.857i)13-s + (0.916 − 0.400i)14-s + (−0.894 + 0.447i)15-s + (−0.843 − 0.536i)16-s + (0.952 − 0.304i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(367\)
Sign: $0.974 - 0.223i$
Analytic conductor: \(1.70434\)
Root analytic conductor: \(1.70434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{367} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 367,\ (0:\ ),\ 0.974 - 0.223i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.903444187 - 0.2150535199i\)
\(L(\frac12)\) \(\approx\) \(1.903444187 - 0.2150535199i\)
\(L(1)\) \(\approx\) \(1.559508325 + 0.1138365079i\)
\(L(1)\) \(\approx\) \(1.559508325 + 0.1138365079i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad367 \( 1 \)
good2 \( 1 + (0.600 + 0.799i)T \)
3 \( 1 + (0.870 - 0.492i)T \)
5 \( 1 + (-0.998 - 0.0514i)T \)
7 \( 1 + (0.229 - 0.973i)T \)
11 \( 1 + (-0.376 - 0.926i)T \)
13 \( 1 + (0.514 - 0.857i)T \)
17 \( 1 + (0.952 - 0.304i)T \)
19 \( 1 + (0.978 + 0.204i)T \)
23 \( 1 + (-0.843 + 0.536i)T \)
29 \( 1 + (0.423 - 0.905i)T \)
31 \( 1 + (-0.279 + 0.960i)T \)
37 \( 1 + (-0.0771 + 0.997i)T \)
41 \( 1 + (-0.376 + 0.926i)T \)
43 \( 1 + (0.978 - 0.204i)T \)
47 \( 1 + (-0.784 + 0.620i)T \)
53 \( 1 + (-0.998 - 0.0514i)T \)
59 \( 1 + (0.128 - 0.991i)T \)
61 \( 1 + (0.916 - 0.400i)T \)
67 \( 1 + (0.128 + 0.991i)T \)
71 \( 1 + (-0.279 - 0.960i)T \)
73 \( 1 + (-0.640 - 0.767i)T \)
79 \( 1 + (-0.894 - 0.447i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.784 + 0.620i)T \)
97 \( 1 + (-0.843 + 0.536i)T \)
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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

−24.47081167977805750844737188363, −23.790210168856094201449473810070, −22.73211960804304316364898668756, −21.975984421432140448309489228318, −21.03999126938748004254266442059, −20.45321030774321254841671355858, −19.600704043709955116550596565960, −18.79201325667372952218151742756, −18.18751347719376523681996554318, −16.15552902117778930748637503138, −15.60615818428913496297035807570, −14.68203565272814944908347587703, −14.17585794712157335889863179896, −12.8160361943378861431308979710, −12.10091308761821649673360504163, −11.22066428275187555501292304582, −10.143962047455604493605965639, −9.22719336210885181994846470268, −8.34759899924611394735697549915, −7.20915855592023683214801444289, −5.57063442337912041918444152364, −4.55592902160434369266297895466, −3.74887453898844443539250148639, −2.74400950164486860340839551509, −1.72197145255139952275087184996, 0.947574119317748353876140814340, 3.242515252568213556152143435565, 3.45475018893875470236717338312, 4.75895180927389624596677096211, 6.06961902191627453165579527153, 7.28392716971659658861181646818, 7.94944053156683877766271533966, 8.34238261374063183134865987196, 9.85794650843012595969849459627, 11.31612459942742354180750532461, 12.25061906038837199620689225458, 13.26308797794916048307721726923, 13.927904777371357659308128587958, 14.65219921601898822783628023778, 15.81315489271776661646641616210, 16.18663864991664870552635597138, 17.54433089176023181659149493456, 18.429273274508796363904197075094, 19.40662164098930690715755011852, 20.45152910736186251007784294861, 20.88882369815858931029809819707, 22.25508086848937483419130312729, 23.41654324251835841132558697104, 23.61954500428769670215150092094, 24.55169838517499534999271310587

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