Properties

Label 2-1350-15.2-c1-0-16
Degree $2$
Conductor $1350$
Sign $-0.525 + 0.850i$
Analytic cond. $10.7798$
Root an. cond. $3.28326$
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)8-s − 5.19i·11-s + (−3.67 + 3.67i)13-s − 1.00·16-s + (2.12 − 2.12i)17-s + 2i·19-s + (3.67 + 3.67i)22-s + (−2.12 − 2.12i)23-s − 5.19i·26-s − 5.19·29-s − 5·31-s + (0.707 − 0.707i)32-s + 3i·34-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.250 + 0.250i)8-s − 1.56i·11-s + (−1.01 + 1.01i)13-s − 0.250·16-s + (0.514 − 0.514i)17-s + 0.458i·19-s + (0.783 + 0.783i)22-s + (−0.442 − 0.442i)23-s − 1.01i·26-s − 0.964·29-s − 0.898·31-s + (0.125 − 0.125i)32-s + 0.514i·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.525 + 0.850i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ -0.525 + 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3909431997\)
\(L(\frac12)\) \(\approx\) \(0.3909431997\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good7 \( 1 + 7iT^{2} \)
11 \( 1 + 5.19iT - 11T^{2} \)
13 \( 1 + (3.67 - 3.67i)T - 13iT^{2} \)
17 \( 1 + (-2.12 + 2.12i)T - 17iT^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (2.12 + 2.12i)T + 23iT^{2} \)
29 \( 1 + 5.19T + 29T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 + 10.3iT - 41T^{2} \)
43 \( 1 + (3.67 - 3.67i)T - 43iT^{2} \)
47 \( 1 + (6.36 - 6.36i)T - 47iT^{2} \)
53 \( 1 + (-8.48 - 8.48i)T + 53iT^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + (7.34 + 7.34i)T + 67iT^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + (7.34 - 7.34i)T - 73iT^{2} \)
79 \( 1 - iT - 79T^{2} \)
83 \( 1 + (8.48 + 8.48i)T + 83iT^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + (7.34 + 7.34i)T + 97iT^{2} \)
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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

−9.209282701101867648788386569364, −8.591486766884373833289572529226, −7.67717365678791168443473078178, −7.02291239816863464658540041358, −6.02018704056091979208444631732, −5.39368684352070710149598983070, −4.25500337972743310225787058996, −3.11186551165872092388329979531, −1.77966917527211038749105900184, −0.18295184168711939073708203378, 1.60458891110859627493327031266, 2.59477042384432548923997772541, 3.71241398545032039715353974247, 4.76399872611051412911377048476, 5.59938804280462075384477024869, 6.94302152955925432723436362494, 7.52671400248297288811801891217, 8.230233332134534134040329508882, 9.329185904703530579665334698230, 9.934808056302268314615606658789

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