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Properties

Label 1-203-203.165-r0-0-0
Degree $1$
Conductor $203$
Sign $-0.139 + 0.990i$
Analytic cond. $0.942728$
Root an. cond. $0.942728$
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  + (0.365 + 0.930i)2-s + (0.955 − 0.294i)3-s + (−0.733 + 0.680i)4-s + (0.365 + 0.930i)5-s + (0.623 + 0.781i)6-s + (−0.900 − 0.433i)8-s + (0.826 − 0.563i)9-s + (−0.733 + 0.680i)10-s + (0.826 + 0.563i)11-s + (−0.5 + 0.866i)12-s + (−0.900 + 0.433i)13-s + (0.623 + 0.781i)15-s + (0.0747 − 0.997i)16-s + (−0.5 − 0.866i)17-s + (0.826 + 0.563i)18-s + (0.955 + 0.294i)19-s + ⋯
L(s)  = 1  + (0.365 + 0.930i)2-s + (0.955 − 0.294i)3-s + (−0.733 + 0.680i)4-s + (0.365 + 0.930i)5-s + (0.623 + 0.781i)6-s + (−0.900 − 0.433i)8-s + (0.826 − 0.563i)9-s + (−0.733 + 0.680i)10-s + (0.826 + 0.563i)11-s + (−0.5 + 0.866i)12-s + (−0.900 + 0.433i)13-s + (0.623 + 0.781i)15-s + (0.0747 − 0.997i)16-s + (−0.5 − 0.866i)17-s + (0.826 + 0.563i)18-s + (0.955 + 0.294i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(203\)    =    \(7 \cdot 29\)
Sign: $-0.139 + 0.990i$
Analytic conductor: \(0.942728\)
Root analytic conductor: \(0.942728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{203} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 203,\ (0:\ ),\ -0.139 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.198234474 + 1.379318961i\)
\(L(\frac12)\) \(\approx\) \(1.198234474 + 1.379318961i\)
\(L(1)\) \(\approx\) \(1.322229692 + 0.8847964435i\)
\(L(1)\) \(\approx\) \(1.322229692 + 0.8847964435i\)

Euler product

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

−26.87872234278536626604600563993, −25.64528499118485682645194541984, −24.3886013827548865698077443994, −24.129764968171470914996187001, −22.23920194767052632000911615331, −21.817607260996722505232857944449, −20.79151537402586720019987373504, −19.870980969052979508426958016084, −19.60548239857212136517438415638, −18.20157056399478465966296904050, −17.05556175427277419557903992235, −15.79082960889670778233328858143, −14.60166253654781646940985484744, −13.85806204420498334468390810506, −12.93628284448142354783621847059, −12.09079355941655809340915408730, −10.70413646323479185694798129667, −9.56876686096894423413174508151, −9.03263669146323464446620901009, −7.89818779769002362683153817271, −5.932442495571910505605048527930, −4.70806865512145152012191723046, −3.80080555656785168603400433802, −2.51196385693746659928677078365, −1.3298508919009449772859894496, 2.146957076945222043602601121487, 3.35543491463115745420839701150, 4.49007872871796839134118489042, 6.09313012187766324176687800291, 7.1433144493107786437532348571, 7.64131278871584599277991246109, 9.264462131427760780622339967874, 9.71062310060543167480831201596, 11.68152419597362544784960368660, 12.773676435686947395736609636428, 14.05249313069074902298713321688, 14.306968593829404080408585087212, 15.23056851030168855148266025283, 16.32130181683618369605516770991, 17.71095590277609056466285187423, 18.252485352965018540035020597718, 19.40218796431327146619696604123, 20.46625241773919655474346267044, 21.795125677628926942898002123285, 22.31853647430287680036681869542, 23.45488047434012368650749705065, 24.67676096149089610433983988547, 25.02491136684590924912238548368, 26.13959491543148339457617453290, 26.62453063972783577997992579712

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