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Properties

Label 2-145-145.57-c0-0-0
Degree $2$
Conductor $145$
Sign $0.850 - 0.525i$
Analytic cond. $0.0723644$
Root an. cond. $0.269006$
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  + i·4-s + i·5-s + (−1 − i)7-s − i·9-s + (1 − i)13-s − 16-s − 20-s + (−1 + i)23-s − 25-s + (1 − i)28-s + i·29-s + (1 − i)35-s + 36-s + 45-s + i·49-s + ⋯
L(s)  = 1  + i·4-s + i·5-s + (−1 − i)7-s − i·9-s + (1 − i)13-s − 16-s − 20-s + (−1 + i)23-s − 25-s + (1 − i)28-s + i·29-s + (1 − i)35-s + 36-s + 45-s + i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(0.0723644\)
Root analytic conductor: \(0.269006\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 145,\ (\ :0),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6005376324\)
\(L(\frac12)\) \(\approx\) \(0.6005376324\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - iT \)
29 \( 1 - iT \)
good2 \( 1 - iT^{2} \)
3 \( 1 + iT^{2} \)
7 \( 1 + (1 + i)T + iT^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-1 + i)T - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (1 - i)T - iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (1 - i)T - iT^{2} \)
59 \( 1 + 2iT - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (-1 - i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-1 + i)T - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.32993598963205229484444286850, −12.54574876897768236408186826010, −11.39422617378668995229584482205, −10.41353880598393309285793247204, −9.403540390534429671056392011072, −7.992400538794846214878210654745, −7.01113802604147820072281225945, −6.16509560494874622589673106473, −3.72658814719173165662651788935, −3.27742719664987583238506859129, 2.07354667645135374752323359783, 4.39077343411487350689405183883, 5.63166093877739375271477810465, 6.42492806225623045654874444485, 8.303183790762553082209274538642, 9.180632172722448789636496637107, 10.01740528623592685061557402512, 11.24964052958724875129278183064, 12.28442879818933150612535536041, 13.34142935627856553675093957670

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