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Properties

Label 2-315-35.27-c1-0-7
Degree $2$
Conductor $315$
Sign $0.437 - 0.899i$
Analytic cond. $2.51528$
Root an. cond. $1.58596$
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  + (1 + i)2-s + (1.58 + 1.58i)5-s + (−0.581 + 2.58i)7-s + (2 − 2i)8-s + 3.16i·10-s + 11-s + (−1.58 − 1.58i)13-s + (−3.16 + 2i)14-s + 4·16-s + (−1.58 + 1.58i)17-s + 3.16·19-s + (1 + i)22-s + (−2 + 2i)23-s + 5.00i·25-s − 3.16i·26-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (0.707 + 0.707i)5-s + (−0.219 + 0.975i)7-s + (0.707 − 0.707i)8-s + 1.00i·10-s + 0.301·11-s + (−0.438 − 0.438i)13-s + (−0.845 + 0.534i)14-s + 16-s + (−0.383 + 0.383i)17-s + 0.725·19-s + (0.213 + 0.213i)22-s + (−0.417 + 0.417i)23-s + 1.00i·25-s − 0.620i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.437 - 0.899i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.437 - 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74454 + 1.09072i\)
\(L(\frac12)\) \(\approx\) \(1.74454 + 1.09072i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-1.58 - 1.58i)T \)
7 \( 1 + (0.581 - 2.58i)T \)
good2 \( 1 + (-1 - i)T + 2iT^{2} \)
11 \( 1 - T + 11T^{2} \)
13 \( 1 + (1.58 + 1.58i)T + 13iT^{2} \)
17 \( 1 + (1.58 - 1.58i)T - 17iT^{2} \)
19 \( 1 - 3.16T + 19T^{2} \)
23 \( 1 + (2 - 2i)T - 23iT^{2} \)
29 \( 1 + 3iT - 29T^{2} \)
31 \( 1 + 3.16iT - 31T^{2} \)
37 \( 1 + (6 + 6i)T + 37iT^{2} \)
41 \( 1 + 9.48iT - 41T^{2} \)
43 \( 1 + (3 - 3i)T - 43iT^{2} \)
47 \( 1 + (-4.74 + 4.74i)T - 47iT^{2} \)
53 \( 1 + (1 - i)T - 53iT^{2} \)
59 \( 1 + 9.48T + 59T^{2} \)
61 \( 1 + 6.32iT - 61T^{2} \)
67 \( 1 + (1 + i)T + 67iT^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 - 13iT - 79T^{2} \)
83 \( 1 + (-3.16 - 3.16i)T + 83iT^{2} \)
89 \( 1 - 6.32T + 89T^{2} \)
97 \( 1 + (1.58 - 1.58i)T - 97iT^{2} \)
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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

−12.01739035434676919289925875985, −10.82519585479477571320196020160, −9.918459669709633565628360605224, −9.149981013027140106365553654337, −7.68130652035388345136863093330, −6.68631560571635321228777315088, −5.85770381043851349133071457343, −5.20694083108049077105125199954, −3.63016684718638810785968938173, −2.13191334659042768959220316744, 1.55090239595375865639954210336, 3.06583101621596069413765010257, 4.34831625014759598932213254230, 5.06324056398501151435485557082, 6.52690765283021155733060995138, 7.63339777848604071473896654626, 8.807421473737900457411843848123, 9.836763660604467706312328341122, 10.65385489279237961667549441439, 11.73842403705828335071219904378

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