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Properties

Label 2-3150-15.8-c1-0-27
Degree $2$
Conductor $3150$
Sign $0.998 + 0.0618i$
Analytic cond. $25.1528$
Root an. cond. $5.01526$
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  + (0.707 + 0.707i)2-s + 1.00i·4-s + (0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s − 5.26i·11-s + (3.16 + 3.16i)13-s + 1.00·14-s − 1.00·16-s + (−3.05 − 3.05i)17-s + (3.72 − 3.72i)22-s + (−4.32 + 4.32i)23-s + 4.46i·26-s + (0.707 + 0.707i)28-s + 9.96·29-s + 1.26·31-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 − 1.58i·11-s + (0.876 + 0.876i)13-s + 0.267·14-s − 0.250·16-s + (−0.741 − 0.741i)17-s + (0.793 − 0.793i)22-s + (−0.901 + 0.901i)23-s + 0.876i·26-s + (0.133 + 0.133i)28-s + 1.84·29-s + 0.227·31-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

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.998 + 0.0618i$
Analytic conductor: \(25.1528\)
Root analytic conductor: \(5.01526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3150} (2843, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ 0.998 + 0.0618i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.455910178\)
\(L(\frac12)\) \(\approx\) \(2.455910178\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-0.707 + 0.707i)T \)
good11 \( 1 + 5.26iT - 11T^{2} \)
13 \( 1 + (-3.16 - 3.16i)T + 13iT^{2} \)
17 \( 1 + (3.05 + 3.05i)T + 17iT^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (4.32 - 4.32i)T - 23iT^{2} \)
29 \( 1 - 9.96T + 29T^{2} \)
31 \( 1 - 1.26T + 31T^{2} \)
37 \( 1 + (-2.93 + 2.93i)T - 37iT^{2} \)
41 \( 1 + 10.6iT - 41T^{2} \)
43 \( 1 + (0.597 + 0.597i)T + 43iT^{2} \)
47 \( 1 + (-3.42 - 3.42i)T + 47iT^{2} \)
53 \( 1 + (-9.88 + 9.88i)T - 53iT^{2} \)
59 \( 1 - 3.12T + 59T^{2} \)
61 \( 1 - 3.05T + 61T^{2} \)
67 \( 1 + (4.59 - 4.59i)T - 67iT^{2} \)
71 \( 1 + 9.23iT - 71T^{2} \)
73 \( 1 + (10.2 + 10.2i)T + 73iT^{2} \)
79 \( 1 + 3.57iT - 79T^{2} \)
83 \( 1 + (-6.21 + 6.21i)T - 83iT^{2} \)
89 \( 1 + 3.61T + 89T^{2} \)
97 \( 1 + (-4.63 + 4.63i)T - 97iT^{2} \)
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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

−8.739546958810518605974732151823, −7.88856540504639974882037268790, −7.07787087976538763520372917699, −6.29707176780921845658479798212, −5.78244369138691479107711376021, −4.82384710225471684365453165851, −3.99607377658258471173073224776, −3.31902969878940883950723883418, −2.17606727749768082125180184637, −0.73100880806499381951908212707, 1.13309695910282967825420048890, 2.19520962180976842924003113648, 2.94636800630101030530047911341, 4.28313121622641622475991583056, 4.48412959469584296031065539378, 5.60676110266927252051899082650, 6.32608715796797043248598156943, 7.02908610534415542727532875507, 8.181079741040241093462864094932, 8.525863513775470649562124668020

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