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Properties

Label 1-368-368.261-r0-0-0
Degree $1$
Conductor $368$
Sign $0.350 - 0.936i$
Analytic cond. $1.70898$
Root an. cond. $1.70898$
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  + (0.755 + 0.654i)3-s + (−0.989 − 0.142i)5-s + (−0.415 − 0.909i)7-s + (0.142 + 0.989i)9-s + (−0.281 − 0.959i)11-s + (−0.909 − 0.415i)13-s + (−0.654 − 0.755i)15-s + (0.841 − 0.540i)17-s + (0.540 − 0.841i)19-s + (0.281 − 0.959i)21-s + (0.959 + 0.281i)25-s + (−0.540 + 0.841i)27-s + (−0.540 − 0.841i)29-s + (−0.654 − 0.755i)31-s + (0.415 − 0.909i)33-s + ⋯
L(s)  = 1  + (0.755 + 0.654i)3-s + (−0.989 − 0.142i)5-s + (−0.415 − 0.909i)7-s + (0.142 + 0.989i)9-s + (−0.281 − 0.959i)11-s + (−0.909 − 0.415i)13-s + (−0.654 − 0.755i)15-s + (0.841 − 0.540i)17-s + (0.540 − 0.841i)19-s + (0.281 − 0.959i)21-s + (0.959 + 0.281i)25-s + (−0.540 + 0.841i)27-s + (−0.540 − 0.841i)29-s + (−0.654 − 0.755i)31-s + (0.415 − 0.909i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $0.350 - 0.936i$
Analytic conductor: \(1.70898\)
Root analytic conductor: \(1.70898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{368} (261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 368,\ (0:\ ),\ 0.350 - 0.936i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8476545222 - 0.5878611405i\)
\(L(\frac12)\) \(\approx\) \(0.8476545222 - 0.5878611405i\)
\(L(1)\) \(\approx\) \(0.9872340158 - 0.1206238560i\)
\(L(1)\) \(\approx\) \(0.9872340158 - 0.1206238560i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.755 + 0.654i)T \)
5 \( 1 + (-0.989 - 0.142i)T \)
7 \( 1 + (-0.415 - 0.909i)T \)
11 \( 1 + (-0.281 - 0.959i)T \)
13 \( 1 + (-0.909 - 0.415i)T \)
17 \( 1 + (0.841 - 0.540i)T \)
19 \( 1 + (0.540 - 0.841i)T \)
29 \( 1 + (-0.540 - 0.841i)T \)
31 \( 1 + (-0.654 - 0.755i)T \)
37 \( 1 + (0.989 - 0.142i)T \)
41 \( 1 + (0.142 - 0.989i)T \)
43 \( 1 + (-0.755 - 0.654i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.909 + 0.415i)T \)
59 \( 1 + (0.909 + 0.415i)T \)
61 \( 1 + (-0.755 + 0.654i)T \)
67 \( 1 + (-0.281 + 0.959i)T \)
71 \( 1 + (0.959 + 0.281i)T \)
73 \( 1 + (-0.841 - 0.540i)T \)
79 \( 1 + (0.415 - 0.909i)T \)
83 \( 1 + (-0.989 + 0.142i)T \)
89 \( 1 + (0.654 - 0.755i)T \)
97 \( 1 + (-0.142 + 0.989i)T \)
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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

−24.96047567128007332721312950543, −23.918465550305030448858569436723, −23.27990592092580174357527532105, −22.29215204959364013082029955143, −21.24307926339758991619750614818, −20.1198847913922941989533147282, −19.6085548719411018899618001993, −18.64428317767147595265373690473, −18.2310255055871436546056046647, −16.7662729477132355598492800806, −15.73999026931583738188380186252, −14.80406046756509167569415631376, −14.44528139853040757601010247827, −12.74067485951869793751443164244, −12.45060905769088820394713590336, −11.579890378140405685237567418670, −10.00661548717425414238471634827, −9.19229579847661667843214828746, −8.04933326441417765034669616133, −7.447971796409461367928933978178, −6.44608639934224849286140332429, −5.046757384057362576186996967640, −3.68899695882348759136880467154, −2.80728202590022417378529889455, −1.63795463878005773989317538801, 0.57379597030063120607046067266, 2.72554315454893032066396790140, 3.52373836904916885916461006039, 4.41547298748812825015736229184, 5.51235582779145508175778046177, 7.36639544434730819668453448986, 7.70057904734944363411212529783, 8.911581052634731318690736972, 9.845700917776492017775375554057, 10.76651702527072030070031186799, 11.67335981598311103737642125289, 12.98428079470737413450440572628, 13.78225829145737637797457795096, 14.73403854429066211559156197456, 15.62644674794972006093358943175, 16.36369885966882406181816828411, 17.04871867089527672502385280232, 18.70373653732781335813011894988, 19.37289388958837542492864439564, 20.15457915788190952669438211934, 20.70387471958750559325949899462, 21.93764411097844520912757417239, 22.62572933264353836917482549883, 23.70781097352507313261370267137, 24.41519869961324818432294349211

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