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Properties

Label 1-145-145.128-r0-0-0
Degree $1$
Conductor $145$
Sign $0.934 - 0.355i$
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  + 2-s − 3-s + 4-s − 6-s − i·7-s + 8-s + 9-s + i·11-s − 12-s − i·13-s − i·14-s + 16-s + 17-s + 18-s − i·19-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s − 6-s − i·7-s + 8-s + 9-s + i·11-s − 12-s − i·13-s − i·14-s + 16-s + 17-s + 18-s − i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.934 - 0.355i$
Analytic conductor: \(0.673377\)
Root analytic conductor: \(0.673377\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (0:\ ),\ 0.934 - 0.355i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.526411005 - 0.2803797537i\)
\(L(\frac12)\) \(\approx\) \(1.526411005 - 0.2803797537i\)
\(L(1)\) \(\approx\) \(1.445877376 - 0.1421279466i\)
\(L(1)\) \(\approx\) \(1.445877376 - 0.1421279466i\)

Euler product

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

−28.575415900258540769070171878842, −27.608611311885133532323631759423, −26.23466724132784587978975245008, −24.85742853991052127018854336503, −24.25679076388511478548809201219, −23.25878321034084280548333274843, −22.37731131504355664204917169322, −21.51372291747184055668055407656, −20.93904169332364768427162130737, −19.11597165039085724449461550869, −18.482464000858932682744469460362, −16.69518655322467477809083356036, −16.327365445046418472746817567576, −15.075517524403959638932259213152, −14.01457590587866034083915492254, −12.703334850359798019100488694926, −11.91275923756736253828454527806, −11.174147059515731578083194644448, −9.880439347391169883791694346751, −8.12238340655343187603806381914, −6.5447175830808968273790855940, −5.84009431590266780401976831588, −4.82720010917748641348470847112, −3.43233880572970114448586214940, −1.76058535081482654558672248219, 1.40107555267379478213052944477, 3.38521264351522945136246879010, 4.65383926034041601073026630363, 5.50150836053284073336478971035, 6.886327969630382692155581992860, 7.558547676080904308594241522415, 10.01732818063003308736840426979, 10.715215028499248906273346095002, 11.88821601792367444428125804610, 12.77029500395329630977375903801, 13.700537149836299065565886274440, 15.05721319979724988861102300850, 15.94130584297147392423052031316, 17.06802709002563636570938338282, 17.77959849512311369916281998332, 19.48777878421408130500621982103, 20.45008214323424230750962793573, 21.43184146972020823105848749063, 22.530312014516157291647615828028, 23.1949714272981043099428408158, 23.801737446415263314627299028331, 25.00185387492234196054944632173, 26.02897984295937491936307082715, 27.48438060800441617218160639788, 28.30322975755651199153291704508

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