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Properties

Label 1-697-697.645-r1-0-0
Degree $1$
Conductor $697$
Sign $0.771 + 0.636i$
Analytic cond. $74.9030$
Root an. cond. $74.9030$
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  + (0.951 − 0.309i)2-s + (0.707 + 0.707i)3-s + (0.809 − 0.587i)4-s + (0.587 + 0.809i)5-s + (0.891 + 0.453i)6-s + (0.891 − 0.453i)7-s + (0.587 − 0.809i)8-s + i·9-s + (0.809 + 0.587i)10-s + (0.156 + 0.987i)11-s + (0.987 + 0.156i)12-s + (0.453 − 0.891i)13-s + (0.707 − 0.707i)14-s + (−0.156 + 0.987i)15-s + (0.309 − 0.951i)16-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)2-s + (0.707 + 0.707i)3-s + (0.809 − 0.587i)4-s + (0.587 + 0.809i)5-s + (0.891 + 0.453i)6-s + (0.891 − 0.453i)7-s + (0.587 − 0.809i)8-s + i·9-s + (0.809 + 0.587i)10-s + (0.156 + 0.987i)11-s + (0.987 + 0.156i)12-s + (0.453 − 0.891i)13-s + (0.707 − 0.707i)14-s + (−0.156 + 0.987i)15-s + (0.309 − 0.951i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 697 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.771 + 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 697 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.771 + 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(697\)    =    \(17 \cdot 41\)
Sign: $0.771 + 0.636i$
Analytic conductor: \(74.9030\)
Root analytic conductor: \(74.9030\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{697} (645, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 697,\ (1:\ ),\ 0.771 + 0.636i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(6.686347304 + 2.404075790i\)
\(L(\frac12)\) \(\approx\) \(6.686347304 + 2.404075790i\)
\(L(1)\) \(\approx\) \(3.006039521 + 0.5077531563i\)
\(L(1)\) \(\approx\) \(3.006039521 + 0.5077531563i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.951 - 0.309i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (0.587 + 0.809i)T \)
7 \( 1 + (0.891 - 0.453i)T \)
11 \( 1 + (0.156 + 0.987i)T \)
13 \( 1 + (0.453 - 0.891i)T \)
19 \( 1 + (0.453 + 0.891i)T \)
23 \( 1 + (0.309 + 0.951i)T \)
29 \( 1 + (-0.987 - 0.156i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 + (0.951 - 0.309i)T \)
47 \( 1 + (-0.891 - 0.453i)T \)
53 \( 1 + (-0.987 - 0.156i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.951 + 0.309i)T \)
67 \( 1 + (0.156 - 0.987i)T \)
71 \( 1 + (-0.156 - 0.987i)T \)
73 \( 1 + iT \)
79 \( 1 + (-0.707 - 0.707i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.891 + 0.453i)T \)
97 \( 1 + (0.156 - 0.987i)T \)
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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

−22.167055489730766629004087533861, −21.51214047710300195052111631032, −20.78602926882424472286218487508, −20.29545580862364005849615051212, −19.19727344175447675582958316127, −18.26400795160300167561181734707, −17.39251429797064524056297853926, −16.496765892509222168126895633160, −15.70315238701187186018138980459, −14.43134232430401220471105044519, −14.2455981826644227201767266771, −13.2406490645749164422013127159, −12.74930476786240773106009792462, −11.67442051309675680104647709446, −11.10530112359061373140087115395, −9.27570543142071467346968727355, −8.64733999097996606090059217371, −7.92707516317162983581031314796, −6.78388926478281431241181353637, −5.98079703071077455260078114882, −5.07595042449029192152006598648, −4.108925163409915208780753588467, −2.91247964868369763596439367014, −1.97176481252767565766453383779, −1.111058532942720267858376644640, 1.52615102941016876170297939175, 2.2554406239127848166145885533, 3.39575512857560767575750862297, 4.02744402196105064843839915014, 5.15657338503768115499627521617, 5.83668267983628717134340711024, 7.30275725181457729232270183489, 7.76944125621898257732676993186, 9.40090666814922889302949161853, 10.08968837199931321521730880159, 10.83850222846497275309184398311, 11.47363696732063334312624910675, 12.883217787947514481374925544629, 13.57064771260506847414327225740, 14.447841611494729787114669779254, 14.82948002639147495951623500995, 15.51333352656431645641278374987, 16.64070824937003715104970126076, 17.689516651103219235121729979472, 18.58377704398415343463685555068, 19.66812740714238808800039046277, 20.47922320329123952318448266203, 20.86547074517203371875761802501, 21.66273981188285652337046449502, 22.54947808816332264801000600832

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