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 − 1.77i·11-s + (−0.692 − 0.692i)13-s − 1.00·14-s − 1.00·16-s + (2.39 + 2.39i)17-s + (1.25 − 1.25i)22-s + (−3.38 + 3.38i)23-s − 0.979i·26-s + (−0.707 − 0.707i)28-s − 4.42·29-s − 5.77·31-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 − 0.536i·11-s + (−0.192 − 0.192i)13-s − 0.267·14-s − 0.250·16-s + (0.580 + 0.580i)17-s + (0.268 − 0.268i)22-s + (−0.705 + 0.705i)23-s − 0.192i·26-s + (−0.133 − 0.133i)28-s − 0.821·29-s − 1.03·31-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  :  $\chi_{3150} (2843, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.998 - 0.0618i)$
$L(1)$  $\approx$  $1.055169580$
$L(\frac12)$  $\approx$  $1.055169580$
$L(\frac{3}{2})$   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 \( 1 + 1.77iT - 11T^{2} \)
13 \( 1 + (0.692 + 0.692i)T + 13iT^{2} \)
17 \( 1 + (-2.39 - 2.39i)T + 17iT^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (3.38 - 3.38i)T - 23iT^{2} \)
29 \( 1 + 4.42T + 29T^{2} \)
31 \( 1 + 5.77T + 31T^{2} \)
37 \( 1 + (5.91 - 5.91i)T - 37iT^{2} \)
41 \( 1 - 0.807iT - 41T^{2} \)
43 \( 1 + (-4.64 - 4.64i)T + 43iT^{2} \)
47 \( 1 + (-7.47 - 7.47i)T + 47iT^{2} \)
53 \( 1 + (3.56 - 3.56i)T - 53iT^{2} \)
59 \( 1 + 5.89T + 59T^{2} \)
61 \( 1 - 2.39T + 61T^{2} \)
67 \( 1 + (-0.641 + 0.641i)T - 67iT^{2} \)
71 \( 1 - 10.6iT - 71T^{2} \)
73 \( 1 + (5.70 + 5.70i)T + 73iT^{2} \)
79 \( 1 + 16.3iT - 79T^{2} \)
83 \( 1 + (-0.171 + 0.171i)T - 83iT^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 + (12.6 - 12.6i)T - 97iT^{2} \)
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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

−9.004238489346823164505799553698, −8.089023242410737937646549549164, −7.59586751322685032033210012530, −6.71967593874478904150510768852, −5.80868388112595234737761174339, −5.54039079057918255100468712464, −4.39428343799327023799886800526, −3.58240181729126700598357497036, −2.83212625491807509768174662860, −1.54561443447422052246043433560, 0.25779115565137003681502248246, 1.71800518527658592948773374683, 2.57550672349662322021312789467, 3.66170734164865048100813005812, 4.22583333035667839992437184089, 5.26364644620158321730673916302, 5.79684467848489195321120502721, 6.94607902601675134229751504413, 7.31478506990992306148149414673, 8.381506284267556653538248464174

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