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Properties

Label 1-145-145.109-r0-0-0
Degree $1$
Conductor $145$
Sign $-0.132 + 0.991i$
Analytic cond. $0.673377$
Root an. cond. $0.673377$
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  + (−0.900 + 0.433i)2-s + (0.623 + 0.781i)3-s + (0.623 − 0.781i)4-s + (−0.900 − 0.433i)6-s + (−0.623 − 0.781i)7-s + (−0.222 + 0.974i)8-s + (−0.222 + 0.974i)9-s + (0.222 + 0.974i)11-s + 12-s + (0.222 + 0.974i)13-s + (0.900 + 0.433i)14-s + (−0.222 − 0.974i)16-s + 17-s + (−0.222 − 0.974i)18-s + (−0.623 + 0.781i)19-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)2-s + (0.623 + 0.781i)3-s + (0.623 − 0.781i)4-s + (−0.900 − 0.433i)6-s + (−0.623 − 0.781i)7-s + (−0.222 + 0.974i)8-s + (−0.222 + 0.974i)9-s + (0.222 + 0.974i)11-s + 12-s + (0.222 + 0.974i)13-s + (0.900 + 0.433i)14-s + (−0.222 − 0.974i)16-s + 17-s + (−0.222 − 0.974i)18-s + (−0.623 + 0.781i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $-0.132 + 0.991i$
Analytic conductor: \(0.673377\)
Root analytic conductor: \(0.673377\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (0:\ ),\ -0.132 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5424834443 + 0.6197295827i\)
\(L(\frac12)\) \(\approx\) \(0.5424834443 + 0.6197295827i\)
\(L(1)\) \(\approx\) \(0.7221314723 + 0.4018353512i\)
\(L(1)\) \(\approx\) \(0.7221314723 + 0.4018353512i\)

Euler product

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

−28.04485600449524517635432762744, −26.955950397701131968543559513, −25.93562880423503903051015197651, −25.17438437581035568725135152415, −24.5374752601117739101642384738, −23.068908754503377858609475267765, −21.70745340978892115021218075905, −20.81076577107940079324857843414, −19.60640220781338100287555908126, −19.05856751652565432419484572021, −18.25565199487245867845255531948, −17.17076873940525010901456694005, −15.94283479207872827028001848601, −14.87276439600260211632297625912, −13.30404633128378573374055291614, −12.54605813632061715013730141914, −11.51097442899213514398028168442, −10.14942906924234071074389142181, −8.88071074022179269587677348899, −8.34038356709800429421181341073, −7.01124788865177812747763827100, −5.92552709656886914273561319968, −3.34331893092716073892108199220, −2.6441960372409024482921965908, −0.95779329546773309961166983432, 1.78035526199279723838104125809, 3.480003772049846450635080041768, 4.85493239158959812892053436326, 6.51791421145976693680140356565, 7.56826268046229078677400411901, 8.74584744817655290702763787193, 9.84427335520081136818627775583, 10.29999832281309025405793657532, 11.754319576467391168871455666073, 13.53393870582667428327363956463, 14.59967401477991444326742415556, 15.420406301199152868964245264180, 16.65738793687803773842771629698, 16.9894031936412606929799970046, 18.7140839089821170134539240453, 19.45848255097001132187229525885, 20.42411805953812400061193162676, 21.1844470130896628808424347454, 22.79244458022020359191877290095, 23.59436191271750405576935696317, 25.13472901748614236392809697939, 25.68638012186445963091840713342, 26.492862110746526367022362772803, 27.353012260405331835157384497228, 28.19328412463249056300594553720

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