Properties

Label 2-371-371.195-c0-0-0
Degree $2$
Conductor $371$
Sign $0.618 - 0.785i$
Analytic cond. $0.185153$
Root an. cond. $0.430294$
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  + (−1.10 + 1.59i)2-s + (−0.982 − 2.59i)4-s + (0.568 − 0.822i)7-s + (3.33 + 0.822i)8-s + (0.120 − 0.992i)9-s + (0.213 + 0.112i)11-s + (0.688 + 1.81i)14-s + (−2.92 + 2.59i)16-s + (1.45 + 1.28i)18-s + (−0.414 + 0.217i)22-s + 1.77·23-s + (−0.970 − 0.239i)25-s + (−2.69 − 0.663i)28-s + (−1.71 + 0.902i)29-s + (−0.499 − 4.11i)32-s + ⋯
L(s)  = 1  + (−1.10 + 1.59i)2-s + (−0.982 − 2.59i)4-s + (0.568 − 0.822i)7-s + (3.33 + 0.822i)8-s + (0.120 − 0.992i)9-s + (0.213 + 0.112i)11-s + (0.688 + 1.81i)14-s + (−2.92 + 2.59i)16-s + (1.45 + 1.28i)18-s + (−0.414 + 0.217i)22-s + 1.77·23-s + (−0.970 − 0.239i)25-s + (−2.69 − 0.663i)28-s + (−1.71 + 0.902i)29-s + (−0.499 − 4.11i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.618 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.618 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(371\)    =    \(7 \cdot 53\)
Sign: $0.618 - 0.785i$
Analytic conductor: \(0.185153\)
Root analytic conductor: \(0.430294\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{371} (195, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 371,\ (\ :0),\ 0.618 - 0.785i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4908761843\)
\(L(\frac12)\) \(\approx\) \(0.4908761843\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.568 + 0.822i)T \)
53 \( 1 + (-0.120 - 0.992i)T \)
good2 \( 1 + (1.10 - 1.59i)T + (-0.354 - 0.935i)T^{2} \)
3 \( 1 + (-0.120 + 0.992i)T^{2} \)
5 \( 1 + (0.970 + 0.239i)T^{2} \)
11 \( 1 + (-0.213 - 0.112i)T + (0.568 + 0.822i)T^{2} \)
13 \( 1 + (0.748 + 0.663i)T^{2} \)
17 \( 1 + (-0.885 + 0.464i)T^{2} \)
19 \( 1 + (0.748 + 0.663i)T^{2} \)
23 \( 1 - 1.77T + T^{2} \)
29 \( 1 + (1.71 - 0.902i)T + (0.568 - 0.822i)T^{2} \)
31 \( 1 + (-0.568 + 0.822i)T^{2} \)
37 \( 1 + (-0.530 + 0.470i)T + (0.120 - 0.992i)T^{2} \)
41 \( 1 + (-0.568 - 0.822i)T^{2} \)
43 \( 1 + (-1.12 - 0.992i)T + (0.120 + 0.992i)T^{2} \)
47 \( 1 + (0.970 - 0.239i)T^{2} \)
59 \( 1 + (0.970 - 0.239i)T^{2} \)
61 \( 1 + (-0.885 - 0.464i)T^{2} \)
67 \( 1 + (0.0854 + 0.225i)T + (-0.748 + 0.663i)T^{2} \)
71 \( 1 + (0.850 + 0.753i)T + (0.120 + 0.992i)T^{2} \)
73 \( 1 + (-0.885 + 0.464i)T^{2} \)
79 \( 1 + (-0.645 - 0.935i)T + (-0.354 + 0.935i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.885 + 0.464i)T^{2} \)
97 \( 1 + (0.970 + 0.239i)T^{2} \)
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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

−11.29528365837766818739156441989, −10.55762236637817263131102664968, −9.435181760754342608736789015246, −9.027749725414668082857967808798, −7.78768675474191657690433811374, −7.21574527615966156620471333891, −6.33731376380385662283226435764, −5.28144281919149885574651713113, −4.10882263662856725984741214040, −1.21483920791398874146588636035, 1.72824744916573519901804430076, 2.70625087826847940274429936770, 4.09692696163911337333726664493, 5.33119413206694661542310281635, 7.36239858282369892172752074168, 8.107941921774576925025561498421, 8.986246644160028144693245340541, 9.653953175467028974332683173150, 10.77085159991824567933217325619, 11.29035070441712563717621524381

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