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Properties

Label 1-7e2-49.25-r0-0-0
Degree $1$
Conductor $49$
Sign $0.942 - 0.335i$
Analytic cond. $0.227555$
Root an. cond. $0.227555$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.826 − 0.563i)2-s + (−0.733 + 0.680i)3-s + (0.365 − 0.930i)4-s + (0.955 + 0.294i)5-s + (−0.222 + 0.974i)6-s + (−0.222 − 0.974i)8-s + (0.0747 − 0.997i)9-s + (0.955 − 0.294i)10-s + (0.0747 + 0.997i)11-s + (0.365 + 0.930i)12-s + (−0.900 − 0.433i)13-s + (−0.900 + 0.433i)15-s + (−0.733 − 0.680i)16-s + (−0.988 − 0.149i)17-s + (−0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (0.826 − 0.563i)2-s + (−0.733 + 0.680i)3-s + (0.365 − 0.930i)4-s + (0.955 + 0.294i)5-s + (−0.222 + 0.974i)6-s + (−0.222 − 0.974i)8-s + (0.0747 − 0.997i)9-s + (0.955 − 0.294i)10-s + (0.0747 + 0.997i)11-s + (0.365 + 0.930i)12-s + (−0.900 − 0.433i)13-s + (−0.900 + 0.433i)15-s + (−0.733 − 0.680i)16-s + (−0.988 − 0.149i)17-s + (−0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.942 - 0.335i$
Analytic conductor: \(0.227555\)
Root analytic conductor: \(0.227555\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 49,\ (0:\ ),\ 0.942 - 0.335i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.122815745 - 0.1938615830i\)
\(L(\frac12)\) \(\approx\) \(1.122815745 - 0.1938615830i\)
\(L(1)\) \(\approx\) \(1.280466911 - 0.1806316226i\)
\(L(1)\) \(\approx\) \(1.280466911 - 0.1806316226i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (0.826 - 0.563i)T \)
3 \( 1 + (-0.733 + 0.680i)T \)
5 \( 1 + (0.955 + 0.294i)T \)
11 \( 1 + (0.0747 + 0.997i)T \)
13 \( 1 + (-0.900 - 0.433i)T \)
17 \( 1 + (-0.988 - 0.149i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.988 + 0.149i)T \)
29 \( 1 + (0.623 - 0.781i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.365 + 0.930i)T \)
41 \( 1 + (-0.222 - 0.974i)T \)
43 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + (0.826 - 0.563i)T \)
53 \( 1 + (0.365 - 0.930i)T \)
59 \( 1 + (0.955 - 0.294i)T \)
61 \( 1 + (0.365 + 0.930i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.623 + 0.781i)T \)
73 \( 1 + (0.826 + 0.563i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.900 + 0.433i)T \)
89 \( 1 + (0.0747 - 0.997i)T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−33.91181740593369290783369780544, −32.84547753665400128967984795879, −31.76872881951831081908138836272, −30.33800838877746614975288374930, −29.474373037088049719356341633936, −28.61335759319650900622284794465, −26.72273894166387383440253122905, −25.337739661909230008336026520459, −24.36005746307786329279093367574, −23.71580821367981595569932653160, −21.988045959991187279434883995742, −21.72961554918514272153972861333, −19.80789257991871975676818409348, −18.007496912436223229873483196186, −17.093602146792523730652118848442, −16.11921480034142001995943515043, −14.23381942366225381251726984000, −13.31569894688231298831225380389, −12.2605184342386236118323180501, −10.87798358591115479113488115584, −8.693319548284248728291819716807, −6.95438298346737261907270838384, −5.956319884684112567375983290985, −4.75403887340930169710031481621, −2.29455115051436058876020397693, 2.24339800929317109496992518107, 4.2263532877540372808585200766, 5.47640029502383725092379230024, 6.65890100171355105377608803012, 9.69962263339120392198709477488, 10.318736691759783264319546253761, 11.76618446952053708685270123667, 12.947518065591294161054323178972, 14.480764046614879481990767539, 15.43997528020018259022189405218, 17.10766479957415847740792639573, 18.22384584387758560590029146188, 20.065815936928411652954614701835, 21.109587981789763952545930366350, 22.176714330660254953078481240760, 22.74698673172678379862502231044, 24.20598491182213247613786569680, 25.54698009659237326018041340098, 27.15709383241467038400905555058, 28.368839807944114467393672477788, 29.219700139528940594552965133898, 30.12109595893963834065689889641, 31.61467828956052643899433273280, 32.74631799844587404996258560454, 33.52325351169627776602221447245

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