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Properties

Label 2-740-740.339-c1-0-19
Degree $2$
Conductor $740$
Sign $-0.0739 - 0.997i$
Analytic cond. $5.90892$
Root an. cond. $2.43082$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + 2i·4-s + (−1 + 2i)5-s + (2 − 2i)8-s + 3·9-s + (3 − i)10-s + (−5 + 5i)13-s − 4·16-s + (3 − 3i)17-s + (−3 − 3i)18-s + (−4 − 2i)20-s + (−3 − 4i)25-s + 10·26-s + (−7 + 7i)29-s + (4 + 4i)32-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + i·4-s + (−0.447 + 0.894i)5-s + (0.707 − 0.707i)8-s + 9-s + (0.948 − 0.316i)10-s + (−1.38 + 1.38i)13-s − 16-s + (0.727 − 0.727i)17-s + (−0.707 − 0.707i)18-s + (−0.894 − 0.447i)20-s + (−0.600 − 0.800i)25-s + 1.96·26-s + (−1.29 + 1.29i)29-s + (0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0739 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0739 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $-0.0739 - 0.997i$
Analytic conductor: \(5.90892\)
Root analytic conductor: \(2.43082\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 740,\ (\ :1/2),\ -0.0739 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.435651 + 0.469156i\)
\(L(\frac12)\) \(\approx\) \(0.435651 + 0.469156i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
5 \( 1 + (1 - 2i)T \)
37 \( 1 + (6 + i)T \)
good3 \( 1 - 3T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (5 - 5i)T - 13iT^{2} \)
17 \( 1 + (-3 + 3i)T - 17iT^{2} \)
19 \( 1 - 19iT^{2} \)
23 \( 1 + 23iT^{2} \)
29 \( 1 + (7 - 7i)T - 29iT^{2} \)
31 \( 1 + 31iT^{2} \)
41 \( 1 - 10iT - 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 14iT - 53T^{2} \)
59 \( 1 - 59iT^{2} \)
61 \( 1 + (1 - i)T - 61iT^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 16T + 73T^{2} \)
79 \( 1 - 79iT^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (3 - 3i)T - 89iT^{2} \)
97 \( 1 + (5 - 5i)T - 97iT^{2} \)
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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

−10.56842578546078363999876526192, −9.690242110983053745755253548764, −9.289402987826110967876534863215, −7.889095465280755077757988479061, −7.22583522155789220202747930495, −6.74457544915381568172940646335, −4.87175736850027046647198865613, −3.92063686956543484103229579875, −2.86446577248565805811790527708, −1.68854438450245211138321709949, 0.41687409387860703088249900587, 1.87730054231416310968622906773, 3.82126891615290865082867114937, 5.01269930283249481078866924221, 5.57618074407860974052111877896, 6.91896595231618922640353417546, 7.78579193860585766106208041316, 8.133446921337492919665139900944, 9.355055197230425687941881543723, 9.932428092662077078314283769445

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