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Properties

Label 2-3150-15.8-c1-0-35
Degree $2$
Conductor $3150$
Sign $-0.0618 + 0.998i$
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 − 0.828i·11-s + (0.585 + 0.585i)13-s + 1.00·14-s − 1.00·16-s + (−4.41 − 4.41i)17-s − 7.07i·19-s + (0.585 − 0.585i)22-s + (0.828 − 0.828i)23-s + 0.828i·26-s + (0.707 + 0.707i)28-s − 8.82·29-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 − 0.249i·11-s + (0.162 + 0.162i)13-s + 0.267·14-s − 0.250·16-s + (−1.07 − 1.07i)17-s − 1.62i·19-s + (0.124 − 0.124i)22-s + (0.172 − 0.172i)23-s + 0.162i·26-s + (0.133 + 0.133i)28-s − 1.63·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0618 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0618 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.0618 + 0.998i$
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: \((2,\ 3150,\ (\ :1/2),\ -0.0618 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.093498099\)
\(L(\frac12)\) \(\approx\) \(1.093498099\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.828iT - 11T^{2} \)
13 \( 1 + (-0.585 - 0.585i)T + 13iT^{2} \)
17 \( 1 + (4.41 + 4.41i)T + 17iT^{2} \)
19 \( 1 + 7.07iT - 19T^{2} \)
23 \( 1 + (-0.828 + 0.828i)T - 23iT^{2} \)
29 \( 1 + 8.82T + 29T^{2} \)
31 \( 1 + 3.17T + 31T^{2} \)
37 \( 1 + (8.07 - 8.07i)T - 37iT^{2} \)
41 \( 1 - 0.828iT - 41T^{2} \)
43 \( 1 + (3.58 + 3.58i)T + 43iT^{2} \)
47 \( 1 + (3.58 + 3.58i)T + 47iT^{2} \)
53 \( 1 + (0.828 - 0.828i)T - 53iT^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 - 9.41T + 61T^{2} \)
67 \( 1 + (-2.41 + 2.41i)T - 67iT^{2} \)
71 \( 1 + 3.75iT - 71T^{2} \)
73 \( 1 + (-6 - 6i)T + 73iT^{2} \)
79 \( 1 + 9.31iT - 79T^{2} \)
83 \( 1 + (1.65 - 1.65i)T - 83iT^{2} \)
89 \( 1 - 9.31T + 89T^{2} \)
97 \( 1 + (0.242 - 0.242i)T - 97iT^{2} \)
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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

−8.558929385444726897884298475057, −7.48332957489927484652708969571, −6.97198067701689922011214508711, −6.36409756639021191481151218073, −5.18648576557117771075563492316, −4.85399360872522262035244460324, −3.84958626550992465447425315272, −2.96959824691094650585990790993, −1.92110788801603383279229507753, −0.25919783864245441109465030651, 1.59737673465717301244039763315, 2.15964778754494656710940988926, 3.55511434978685370896416655145, 3.96231241477006061428735349081, 5.05883130479400964215923984459, 5.73237217057801842970653216053, 6.41817635329708063301381044571, 7.39348000535900896828816969266, 8.178711415906555881275734726040, 8.943005044828751082316348725286

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