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Properties

Label 2-3150-15.2-c1-0-28
Degree $2$
Conductor $3150$
Sign $0.391 + 0.920i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.391 + 0.920i$
Analytic conductor: \(25.1528\)
Root analytic conductor: \(5.01526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3150} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ 0.391 + 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.260842538\)
\(L(\frac12)\) \(\approx\) \(1.260842538\)
\(L(\frac{3}{2})\) 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

−8.507032580812715091976685396234, −7.85987033865159843997928622423, −7.23663897861376192332364795613, −6.12720486638442548418589441470, −5.65958324252434898214723199098, −5.03796669401659709873727853716, −3.60083582505337136073993184971, −3.03654162561147083392895078203, −1.53241031809631857285470866917, −0.49979045000293173616935033280, 1.31341415813654528388191711476, 1.97322765638374896370438579363, 3.16706535966035647575700633929, 4.17098866014400906563720710597, 4.66674527167308319110596872169, 5.87791088403959053391260149528, 6.75793109331149619821611741643, 7.44853236416417467655846382784, 8.089145664717877904730457040445, 8.980089353003098925096848127063

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