Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.998 - 0.0618i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s + 2.14i·11-s + (−2.16 + 2.16i)13-s + 1.00·14-s − 1.00·16-s + (4.46 − 4.46i)17-s + (1.51 + 1.51i)22-s + (6.32 + 6.32i)23-s + 3.05i·26-s + (0.707 − 0.707i)28-s − 8.20·29-s − 1.85·31-s + (−0.707 + 0.707i)32-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.267 + 0.267i)7-s + (−0.250 − 0.250i)8-s + 0.647i·11-s + (−0.599 + 0.599i)13-s + 0.267·14-s − 0.250·16-s + (1.08 − 1.08i)17-s + (0.323 + 0.323i)22-s + (1.31 + 1.31i)23-s + 0.599i·26-s + (0.133 − 0.133i)28-s − 1.52·29-s − 0.332·31-s + (−0.125 + 0.125i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.998 - 0.0618i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1457, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.998 - 0.0618i)$
$L(1)$  $\approx$  $2.399530177$
$L(\frac12)$  $\approx$  $2.399530177$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-0.707 - 0.707i)T \)
good11 \( 1 - 2.14iT - 11T^{2} \)
13 \( 1 + (2.16 - 2.16i)T - 13iT^{2} \)
17 \( 1 + (-4.46 + 4.46i)T - 17iT^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (-6.32 - 6.32i)T + 23iT^{2} \)
29 \( 1 + 8.20T + 29T^{2} \)
31 \( 1 + 1.85T + 31T^{2} \)
37 \( 1 + (-5.13 - 5.13i)T + 37iT^{2} \)
41 \( 1 - 7.56iT - 41T^{2} \)
43 \( 1 + (-7.84 + 7.84i)T - 43iT^{2} \)
47 \( 1 + (5.01 - 5.01i)T - 47iT^{2} \)
53 \( 1 + (-2.35 - 2.35i)T + 53iT^{2} \)
59 \( 1 - 9.35T + 59T^{2} \)
61 \( 1 + 4.46T + 61T^{2} \)
67 \( 1 + (-3.84 - 3.84i)T + 67iT^{2} \)
71 \( 1 - 0.420iT - 71T^{2} \)
73 \( 1 + (-2.63 + 2.63i)T - 73iT^{2} \)
79 \( 1 + 5.23iT - 79T^{2} \)
83 \( 1 + (-10.6 - 10.6i)T + 83iT^{2} \)
89 \( 1 - 14.5T + 89T^{2} \)
97 \( 1 + (-0.606 - 0.606i)T + 97iT^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.013718315815437440825702168393, −7.62109288021923122574017038466, −7.37723799716314148329228558171, −6.33495017672180265565162812674, −5.28581808829466715393522522990, −4.99108625820172357406749940440, −3.94326685021646214639370664395, −3.05337430997156636288091371999, −2.16518997249384665978596856208, −1.12520030140201723699817456950, 0.71406542007926834295524987654, 2.20954131877944094732523088243, 3.28438097186458471600921194615, 3.96309782515593077645476213269, 4.97139580081825624594661180254, 5.61705771207942796330616677631, 6.28980480200662486608576300590, 7.28059514408427202296642704864, 7.76910578921765018474912773564, 8.519235454547818037394477700795

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