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Properties

Label 2-3150-15.8-c1-0-7
Degree $2$
Conductor $3150$
Sign $-0.296 - 0.955i$
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 + 3.41i·11-s + (2.02 + 2.02i)13-s + 1.00·14-s − 1.00·16-s + (4.37 + 4.37i)17-s + 4.87i·19-s + (2.41 − 2.41i)22-s + (3.74 − 3.74i)23-s − 2.86i·26-s + (−0.707 − 0.707i)28-s − 5.21·29-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 + 1.02i·11-s + (0.561 + 0.561i)13-s + 0.267·14-s − 0.250·16-s + (1.06 + 1.06i)17-s + 1.11i·19-s + (0.514 − 0.514i)22-s + (0.780 − 0.780i)23-s − 0.561i·26-s + (−0.133 − 0.133i)28-s − 0.968·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.296 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.296 - 0.955i$
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: \((2,\ 3150,\ (\ :1/2),\ -0.296 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8946183133\)
\(L(\frac12)\) \(\approx\) \(0.8946183133\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.41iT - 11T^{2} \)
13 \( 1 + (-2.02 - 2.02i)T + 13iT^{2} \)
17 \( 1 + (-4.37 - 4.37i)T + 17iT^{2} \)
19 \( 1 - 4.87iT - 19T^{2} \)
23 \( 1 + (-3.74 + 3.74i)T - 23iT^{2} \)
29 \( 1 + 5.21T + 29T^{2} \)
31 \( 1 + 4.93T + 31T^{2} \)
37 \( 1 + (6.87 - 6.87i)T - 37iT^{2} \)
41 \( 1 + 8.77iT - 41T^{2} \)
43 \( 1 + (0.174 + 0.174i)T + 43iT^{2} \)
47 \( 1 + (3.42 + 3.42i)T + 47iT^{2} \)
53 \( 1 + (-6.43 + 6.43i)T - 53iT^{2} \)
59 \( 1 + 2.22T + 59T^{2} \)
61 \( 1 + 15.2T + 61T^{2} \)
67 \( 1 + (-3.81 + 3.81i)T - 67iT^{2} \)
71 \( 1 - 10.9iT - 71T^{2} \)
73 \( 1 + (2.51 + 2.51i)T + 73iT^{2} \)
79 \( 1 - 9.98iT - 79T^{2} \)
83 \( 1 + (-2.35 + 2.35i)T - 83iT^{2} \)
89 \( 1 + 4.07T + 89T^{2} \)
97 \( 1 + (5.64 - 5.64i)T - 97iT^{2} \)
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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

−8.897886542963172876184873207801, −8.341827288881706921444523041994, −7.48429947142283867065144584263, −6.80256554549386248685788836554, −5.91535520577755907880499653164, −5.05610121808052133128197639099, −3.92088373066520918641975095538, −3.41750611450889080586466206106, −2.09973434608755725744010237350, −1.42907564829153804736254489371, 0.34816314193741633628689484018, 1.36033717764353670725704665031, 2.94003930024234042377523537134, 3.51889158290718811334210842458, 4.83242319190297421759591664210, 5.58336381727028345054895400895, 6.14726570016602229207154264889, 7.28642440518033939467759772150, 7.48556772420410923576484910725, 8.519774677046069128333661792592

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