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Properties

Label 1-145-145.39-r1-0-0
Degree $1$
Conductor $145$
Sign $0.947 - 0.318i$
Analytic cond. $15.5824$
Root an. cond. $15.5824$
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  + (−0.433 + 0.900i)2-s + (−0.781 − 0.623i)3-s + (−0.623 − 0.781i)4-s + (0.900 − 0.433i)6-s + (−0.623 + 0.781i)7-s + (0.974 − 0.222i)8-s + (0.222 + 0.974i)9-s + (−0.974 − 0.222i)11-s + i·12-s + (−0.222 + 0.974i)13-s + (−0.433 − 0.900i)14-s + (−0.222 + 0.974i)16-s − i·17-s + (−0.974 − 0.222i)18-s + (−0.781 + 0.623i)19-s + ⋯
L(s)  = 1  + (−0.433 + 0.900i)2-s + (−0.781 − 0.623i)3-s + (−0.623 − 0.781i)4-s + (0.900 − 0.433i)6-s + (−0.623 + 0.781i)7-s + (0.974 − 0.222i)8-s + (0.222 + 0.974i)9-s + (−0.974 − 0.222i)11-s + i·12-s + (−0.222 + 0.974i)13-s + (−0.433 − 0.900i)14-s + (−0.222 + 0.974i)16-s − i·17-s + (−0.974 − 0.222i)18-s + (−0.781 + 0.623i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.947 - 0.318i$
Analytic conductor: \(15.5824\)
Root analytic conductor: \(15.5824\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (1:\ ),\ 0.947 - 0.318i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5622852516 - 0.09197740136i\)
\(L(\frac12)\) \(\approx\) \(0.5622852516 - 0.09197740136i\)
\(L(1)\) \(\approx\) \(0.5311601160 + 0.1118225831i\)
\(L(1)\) \(\approx\) \(0.5311601160 + 0.1118225831i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.433 + 0.900i)T \)
3 \( 1 + (-0.781 - 0.623i)T \)
7 \( 1 + (-0.623 + 0.781i)T \)
11 \( 1 + (-0.974 - 0.222i)T \)
13 \( 1 + (-0.222 + 0.974i)T \)
17 \( 1 - iT \)
19 \( 1 + (-0.781 + 0.623i)T \)
23 \( 1 + (0.900 - 0.433i)T \)
31 \( 1 + (0.433 - 0.900i)T \)
37 \( 1 + (0.974 - 0.222i)T \)
41 \( 1 - iT \)
43 \( 1 + (0.433 + 0.900i)T \)
47 \( 1 + (-0.974 - 0.222i)T \)
53 \( 1 + (0.900 + 0.433i)T \)
59 \( 1 + T \)
61 \( 1 + (0.781 + 0.623i)T \)
67 \( 1 + (-0.222 - 0.974i)T \)
71 \( 1 + (0.222 - 0.974i)T \)
73 \( 1 + (-0.433 - 0.900i)T \)
79 \( 1 + (0.974 - 0.222i)T \)
83 \( 1 + (-0.623 - 0.781i)T \)
89 \( 1 + (0.433 - 0.900i)T \)
97 \( 1 + (-0.781 + 0.623i)T \)
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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

−28.172261615434731948103188763431, −27.1999487482441088986624095801, −26.414447607625467231182185186467, −25.56470510575041146070119345109, −23.56108169950782979595094677556, −23.00740679750947008669616744712, −21.93648492418849675617832774353, −21.10942552009785325030557593654, −20.14414660931025877251161313870, −19.17987507871032376896460968590, −17.81764625432699667731793431793, −17.22845419089955173569608508948, −16.189191891187962882789819938322, −15.02774301099522879862673069451, −13.15617518435185485738551280797, −12.667562130664266322474569959912, −11.17205667564288504322458306262, −10.43116563207435067338445980503, −9.770586859047025939854242156704, −8.30855970879941938386213635381, −6.90457117872441308611591072733, −5.25575635140688759182704730370, −4.087404709235010389257820244477, −2.899656063163035255435588528269, −0.82196142564838956574628946333, 0.4131032353896725416936010543, 2.28185647949238071742961305747, 4.70995747047288833867801155250, 5.77562852738420511350587721660, 6.65192259653367850341229858030, 7.71761443172507988341374022901, 8.95397811350357417384607625047, 10.141427150214755158903945698678, 11.39649001558167132227207241903, 12.700827381300463657227165516785, 13.59751268063440116620634867966, 14.95769495895574219339864001812, 16.16499116341413579870109612975, 16.67210476590795944634020395658, 17.97666126043691394217632993381, 18.736540569642264533857461556728, 19.29122647747906152622570369528, 21.20272464128484944182351429556, 22.45606606176579826461840142563, 23.15465217497840021815762971133, 24.12194471466443413248770252561, 24.92651614297681355643006454118, 25.81691688324162844985446845572, 26.91832125492974541216719254209, 28.00850205511878260167577847873

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